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origin : the point where the two axes cross in the center of the plane, described by the ordered pair(0,0)(0,0) | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
perimeter : in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle:P=2L+2WP=2L+2W | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
polynomial equation : an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
Pythagorean Theorem : a theorem that states the relationship among the lengths of the sides of a right triangle, used to solve right triangle problems | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
quadrant : one quarter of the coordinate plane, created when the axes divide the plane into four sections | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
quadratic equation : an equation containing a second-degree polynomial; can be solved using multiple methods | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
quadratic formula : a formula that will solve all quadratic equations | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
radical equation : an equation containing at least one radical term where the variable is part of the radicand | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
rational equation : an equation consisting of a fraction of polynomials | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
slope : the change iny-values over the change inx-values | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
solution set : the set of all solutions to an equation | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
square root property : one of the methods used to solve a quadratic equation, in which thex2x2term is isolated so that the square root of both sides of the equation can be taken to solve forx | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
volume : in cubic units, the volume measurement includes length, width, and depth:V=LWHV=LWH | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
x-axis : the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
x-coordinate : the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
x-intercept : the point where a graph intersects thex-axis; an ordered pair with ay-coordinate of zero | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
y-axis : the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
y-coordinate : the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
y-intercept : a point where a graph intercepts they-axis; an ordered pair with anx-coordinate of zero | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
zero-product property : the property that formally states that multiplication by zero is zero, so that each factor of a quadratic equation can be set equal to zero to solve equations | https://openstax.org/books/college-algebra-2e/pages/2-key-terms |
A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. SeeExample 1andExample 2. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Function notation is a shorthand method for relating the input to the output in the formy=f(x).y=f(x).SeeExample 3andExample 4. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
In tabular form, a function can be represented by rows or columns that relate to input and output values. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
To solve for a specific function value, we determine the input values that yield the specific output value. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
An algebraic form of a function can be written from an equation. SeeExample 9andExample 10. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Input and output values of a function can be identified from a table. SeeExample 11. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Relating input values to output values on a graph is another way to evaluate a function. SeeExample 12. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function is one-to-one if each output value corresponds to only one input value. SeeExample 13. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. SeeExample 14. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The graph of a one-to-one function passes the horizontal line test. SeeExample 15. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The domain of a function can be determined by listing the input values of a set of ordered pairs. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The domain of a function can also be determined by identifying the input values of a function written as an equation. SeeExample 2,Example 3, andExample 4. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
For many functions, the domain and range can be determined from a graph. SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
An understanding of toolkit functions can be used to find the domain and range of related functions. SeeExample 8,Example 9, andExample 10. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A piecewise function is described by more than one formula. SeeExample 11andExample 12. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A piecewise function can be graphed using each algebraic formula on its assigned subdomain. SeeExample 13. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Identifying points that mark the interval on a graph can be used to find the average rate of change. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Comparing pairs of input and output values in a table can also be used to find the average rate of change. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. SeeExample 4andExample 5. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The average rate of change can sometimes be determined as an expression. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Minima and maxima are also called extrema. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
We can find local extrema from a graph. SeeExample 8andExample 9. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The highest and lowest points on a graph indicate the maxima and minima. SeeExample 10. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
We can perform algebraic operations on functions. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
When functions are composed, the output of the first (inner) function becomes the input of the second (outer) function. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The function produced by composing two functions is a composite function. SeeExample 2andExample 3. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The order of function composition must be considered when interpreting the meaning of composite functions. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A composite function can be evaluated from a table. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A composite function can be evaluated from a graph. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A composite function can be evaluated from a formula. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. SeeExample 8andExample 9. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Functions can often be decomposed in more than one way. SeeExample 10. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function can be shifted vertically by adding a constant to the output. SeeExample 1andExample 2. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function can be shifted horizontally by adding a constant to the input. SeeExample 3,Example 4, andExample 5. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Vertical and horizontal shifts are often combined. SeeExample 7andExample 8. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A vertical reflection reflects a graph about thex-x-axis. A graph can be reflected vertically by multiplying the output by â1. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A horizontal reflection reflects a graph about they-y-axis. A graph can be reflected horizontally by multiplying the input by â1. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. SeeExample 10. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function presented as an equation can be reflected by applying transformations one at a time. SeeExample 11. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Even functions are symmetric about they-y-axis, whereas odd functions are symmetric about the origin. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Even functions satisfy the conditionf(x)=f(âx).f(x)=f(âx). | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Odd functions satisfy the conditionf(x)=âf(âx).f(x)=âf(âx). | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function can be odd, even, or neither. SeeExample 12. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function can be compressed or stretched vertically by multiplying the output by a constant. SeeExample 13,Example 14, andExample 15. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
A function can be compressed or stretched horizontally by multiplying the input by a constant. SeeExample 16,Example 17, andExample 18. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. SeeExample 19andExample 20. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
Applied problems, such as ranges of possible values, can also be solved using the absolute value function. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
In an absolute value equation, an unknown variable is the input of an absolute value function. | https://openstax.org/books/college-algebra-2e/pages/3-key-concepts |
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/college-algebra-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/college-algebra-2e/pages/3-key-concepts |
Only some of the toolkit functions have an inverse. SeeExample 4. | https://openstax.org/books/college-algebra-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/college-algebra-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/college-algebra-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/college-algebra-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/college-algebra-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/college-algebra-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/college-algebra-2e/pages/3-key-concepts |
f ( x ) = c , f ( x ) = c , where c c is a constant | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = x f ( x ) = x | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = | x | f ( x ) = | x | | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = x 2 f ( x ) = x 2 | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = x 3 f ( x ) = x 3 | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = 1 x f ( x ) = 1 x | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = 1 x 2 f ( x ) = 1 x 2 | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = x f ( x ) = x | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
f ( x ) = x 3 f ( x ) = x 3 | https://openstax.org/books/college-algebra-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/college-algebra-2e/pages/3-key-equations |
( f â g ) ( x ) = f ( g ( x ) ) ( f â g ) ( x ) = f ( g ( x ) ) | https://openstax.org/books/college-algebra-2e/pages/3-key-equations |
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