Learn on PengiCalifornia Reveal Math, Algebra 1Unit 4: Creating Linear Equations

4-5 Inverses of Linear Functions

In this Grade 9 lesson from California Reveal Math, Algebra 1 (Unit 4), students learn how to find and graph inverse relations and inverse functions of linear equations. The lesson covers exchanging x- and y-coordinates to determine inverse relations, reflecting graphs over the line y = x, and solving for f⁻¹(x) by interchanging variables and isolating y. Students also apply inverse linear functions to real-world problems, such as calculating rental time from a total cost function.

Section 1

Inverse relation

The inverse relation is the set of ordered pairs obtained by reversing the coordinates in each ordered pair of a relation rr. So if (a,b)(a, b) is in relation rr, then (b,a)(b, a) is in the inverse relation. The inverse may or may not be a function.

The inverse of the relation {(-4, 8), (0, 2), (3, 2)} is {(8, -4), (2, 0), (2, 3)}.: If a function's graph contains the point (−3,9)(-3, 9), its inverse relation must contain the point (9,−3)(9, -3).: Graphically, a point (a,b)(a, b) and its inverse (b,a)(b, a) are perfect reflections of each other across the line y=xy=x.

Imagine your coordinates are wearing shoes on the wrong feet! To find the inverse, you just swap them. The x-value becomes the y-value, and the y-value becomes the x-value. If a point is (5,1)(5, 1), its inverse buddy is (1,5)(1, 5). This simple switcheroo gives you the inverse for every single point in the relation.

Section 2

Inverse Function Notation: f⁻¹(x) Is Not a Reciprocal

Property

The notation f−1(x)f^{-1}(x) means the inverse function of ff, NOT the reciprocal of f(x)f(x).

f−1(x)≠1f(x)f^{-1}(x) \neq \frac{1}{f(x)}

Section 3

Graphing Inverse Functions by Reflection Across y = x

Property

To graph an inverse function f−1(x)f^{-1}(x), reflect the graph of f(x)f(x) across the line y=xy = x by swapping the xx and yy coordinates of each point. If (a,b)(a, b) is on the graph of f(x)f(x), then (b,a)(b, a) is on the graph of f−1(x)f^{-1}(x).

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Inverse relation

The inverse relation is the set of ordered pairs obtained by reversing the coordinates in each ordered pair of a relation rr. So if (a,b)(a, b) is in relation rr, then (b,a)(b, a) is in the inverse relation. The inverse may or may not be a function.

The inverse of the relation {(-4, 8), (0, 2), (3, 2)} is {(8, -4), (2, 0), (2, 3)}.: If a function's graph contains the point (−3,9)(-3, 9), its inverse relation must contain the point (9,−3)(9, -3).: Graphically, a point (a,b)(a, b) and its inverse (b,a)(b, a) are perfect reflections of each other across the line y=xy=x.

Imagine your coordinates are wearing shoes on the wrong feet! To find the inverse, you just swap them. The x-value becomes the y-value, and the y-value becomes the x-value. If a point is (5,1)(5, 1), its inverse buddy is (1,5)(1, 5). This simple switcheroo gives you the inverse for every single point in the relation.

Section 2

Inverse Function Notation: f⁻¹(x) Is Not a Reciprocal

Property

The notation f−1(x)f^{-1}(x) means the inverse function of ff, NOT the reciprocal of f(x)f(x).

f−1(x)≠1f(x)f^{-1}(x) \neq \frac{1}{f(x)}

Section 3

Graphing Inverse Functions by Reflection Across y = x

Property

To graph an inverse function f−1(x)f^{-1}(x), reflect the graph of f(x)f(x) across the line y=xy = x by swapping the xx and yy coordinates of each point. If (a,b)(a, b) is on the graph of f(x)f(x), then (b,a)(b, a) is on the graph of f−1(x)f^{-1}(x).

Examples