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If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
Ifg(x)g(x)is the inverse off(x),f(x),theng(f(x))=f(g(x))=x.g(f(x))=f(g(x))=x.SeeExample 1,Example 2, andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
Only some of the toolkit functions have an inverse. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
For a function to have an inverse, it must be one-to-one (pass the horizontal line test). | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
A function that is not one-to-one over its entire domain may be one-to-one on part of its domain. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
For a tabular function, exchange the input and output rows to obtain the inverse. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
The inverse of a function can be determined at specific points on its graph. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
To find the inverse of a formula, solve the equationy=f(x)y=f(x)forxxas a function ofy.y.Then exchange the labelsxxandy.y.SeeExample 7,Example 8, andExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
The graph of an inverse function is the reflection of the graph of the original function across the liney=x.y=x.SeeExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-concepts |
f ( x ) = c , f ( x ) = c , where c c is a constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = x f ( x ) = x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = | x | f ( x ) = | x | | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = x 2 f ( x ) = x 2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = x 3 f ( x ) = x 3 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = 1 x f ( x ) = 1 x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = 1 x 2 f ( x ) = 1 x 2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = x f ( x ) = x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
f ( x ) = x 3 f ( x ) = x 3 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
Î y Î x = f ( x 2 ) â f ( x 1 ) x 2 â x 1 Î y Î x = f ( x 2 ) â f ( x 1 ) x 2 â x 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
( f â g ) ( x ) = f ( g ( x ) ) ( f â g ) ( x ) = f ( g ( x ) ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = f ( x ) + k g ( x ) = f ( x ) + k (up for k > 0 k > 0 ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = f ( x â h ) g ( x ) = f ( x â h ) (right for h > 0 h > 0 ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = â f ( x ) g ( x ) = â f ( x ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = f ( â x ) g ( x ) = f ( â x ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = a f ( x ) g ( x ) = a f ( x ) ( a > 0 a > 0 ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = a f ( x ) g ( x ) = a f ( x ) ( 0 < a < 1 ) ( 0 < a < 1 ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = f ( b x ) g ( x ) = f ( b x ) ( 0 < b < 1 ) ( 0 < b < 1 ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
g ( x ) = f ( b x ) g ( x ) = f ( b x ) ( b > 1 b > 1 ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-equations |
absolute maximum : the greatest value of a function over an interval | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
absolute minimum : the lowest value of a function over an interval | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
average rate of change : the difference in the output values of a function found for two values of the input divided by the difference between the inputs | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
composite function : the new function formed by function composition, when the output of one function is used as the input of another | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
decreasing function : a function is decreasing in some open interval iff(b)<f(a)f(b)<f(a)for any two input valuesaaandbbin the given interval whereb>ab>a | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
dependent variable : an output variable | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
domain : the set of all possible input values for a relation | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
even function : a function whose graph is unchanged by horizontal reflection,f(x)=f(âx),f(x)=f(âx),and is symmetric about they-y-axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
function : a relation in which each input value yields a unique output value | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
horizontal compression : a transformation that compresses a functionâs graph horizontally, by multiplying the input by a constantb>1b>1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
horizontal line test : a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
horizontal reflection : a transformation that reflects a functionâs graph across they-axis by multiplying the input byâ1â1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
horizontal shift : a transformation that shifts a functionâs graph left or right by adding a positive or negative constant to the input | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
horizontal stretch : a transformation that stretches a functionâs graph horizontally by multiplying the input by a constant0<b<10<b<1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
increasing function : a function is increasing in some open interval iff(b)>f(a)f(b)>f(a)for any two input valuesaaandbbin the given interval whereb>ab>a | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
independent variable : an input variable | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
input : each object or value in a domain that relates to another object or value by a relationship known as a function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
interval notation : a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
inverse function : for any one-to-one functionf(x),f(x),the inverse is a functionfâ1(x)fâ1(x)such thatfâ1(f(x))=xfâ1(f(x))=xfor allxxin the domain off;f;this also implies thatf(fâ1(x))=xf(fâ1(x))=xfor allxxin the domain offâ1fâ1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
local extrema : collectively, all of a function's local maxima and minima | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
local maximum : a value of the input where a function changes from increasing to decreasing as the input value increases. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
local minimum : a value of the input where a function changes from decreasing to increasing as the input value increases. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
odd function : a function whose graph is unchanged by combined horizontal and vertical reflection,f(x)=âf(âx),f(x)=âf(âx),and is symmetric about the origin | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
one-to-one function : a function for which each value of the output is associated with a unique input value | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
output : each object or value in the range that is produced when an input value is entered into a function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
piecewise function : a function in which more than one formula is used to define the output | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
range : the set of output values that result from the input values in a relation | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
rate of change : the change of an output quantity relative to the change of the input quantity | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
relation : a set of ordered pairs | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
set-builder notation : a method of describing a set by a rule that all of its members obey; it takes the form{x|statement aboutx}{x|statement aboutx} | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
vertical compression : a function transformation that compresses the functionâs graph vertically by multiplying the output by a constant0<a<10<a<1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
vertical line test : a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
vertical reflection : a transformation that reflects a functionâs graph across thex-axis by multiplying the output byâ1â1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
vertical shift : a transformation that shifts a functionâs graph up or down by adding a positive or negative constant to the output | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
vertical stretch : a transformation that stretches a functionâs graph vertically by multiplying the output by a constanta>1a>1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-key-terms |
Linear functions can be represented in words, function notation, tabular form, and graphical form. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Slope is a rate of change. The slope of a linear function can be calculated by dividing the difference betweeny-values by the difference in correspondingx-values of any two points on the line. SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
An equation for a linear function can be written from a graph. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
The equation for a linear function can be written if the slopemmand initial valuebbare known. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
A linear function can be used to solve real-world problems given information in different forms. SeeExample 8,Example 9,andExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Linear functions can be graphed by plotting points or by using they-intercept and slope. SeeExample 11andExample 12. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. SeeExample 13. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
The equation for a linear function can be written by interpreting the graph. SeeExample 14. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Thex-intercept is the point at which the graph of a linear function crosses thex-axis. SeeExample 15. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Horizontal lines are written in the form,f(x)=b.f(x)=b.SeeExample 16. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Vertical lines are written in the form,x=b.x=b.SeeExample 17. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. SeeExample 18. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and thex- andy-values of the given point into the equation,f(x)=mx+b,f(x)=mx+b,and using thebbthat results. Similarly, the point-slope form of an equation can also be used. SeeExample 19. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. SeeExample 20andExample 21. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
We can use the same problem strategies that we would use for any type of function. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
When modeling and solving a problem, identify the variables and look for key values, including the slope andy-intercept. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Draw a diagram, where appropriate. SeeExample 2andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Check for reasonableness of the answer. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Linear models may be built by identifying or calculating the slope and using they-intercept.Thex-intercept may be found by settingy=0,y=0,which is setting the expressionmx+bmx+bequal to 0.The point of intersection of a system of linear equations is the point where thex- andy-values are the same. SeeExample 4.A graph of... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Thex-intercept may be found by settingy=0,y=0,which is setting the expressionmx+bmx+bequal to 0. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
The point of intersection of a system of linear equations is the point where thex- andy-values are the same. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
A graph of the system may be used to identify the points where one line falls below (or above) the other line. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Scatter plots show the relationship between two sets of data. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Scatter plots may represent linear or non-linear models. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
The line of best fit may be estimated or calculated, using a calculator or statistical software. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
The correlation coefficient,r,r,indicates the degree of linear relationship between data. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
A regression line best fits the data. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-concepts |
correlation coefficient : a value,r,r,between â1 and 1 that indicates the degree of linear correlation of variables, or how closely a regression line fits a data set. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
decreasing linear function : a function with a negative slope: Iff(x)=mx+b,thenm<0f(x)=mx+b,thenm<0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
extrapolation : predicting a value outside the domain and range of the data | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
horizontal line : a line defined byf(x)=b,f(x)=b,wherebbis a real number. The slope of a horizontal line is 0. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
increasing linear function : a function with a positive slope: Iff(x)=mx+b,thenm>0.f(x)=mx+b,thenm>0. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
interpolation : predicting a value inside the domain and range of the data | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
least squares regression : a statistical technique for fitting a line to data in a way that minimizes the differences between the line and data values | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
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