Grade by Grade
A. Characteristics of Functions · Functions and Equivalent Expressions

Finding the Equation of an Inverse: Linear Functions

A1.7 A1.4 A1.6About 65 minutes
Learning goals
  • I can find the equation of the inverse of a linear function by reversing operations and by swapping x and y.
  • I can write an inverse using f⁻¹(x) notation and check it.
  • I can describe patterns that connect a linear function and its inverse (slopes, intercepts, domain and range).
  • I can find and use inverses in real-world contexts.

Words to know

inverse function f⁻¹
The function that undoes f: if f(a) = b then f⁻¹(b) = a.
swap-and-solve method
To find an inverse: write y = f(x), swap x and y, solve for y.
reciprocal
For a number m ≠ 0, the number 1/m. Inverse lines have reciprocal slopes.
self-inverse
A function that is its own inverse, such as f(x) = −x + 6.

From a picture to a formula

Last lesson you found inverses by swapping table columns and reflecting graphs. That works, but it only gives you some points. A formula for the inverse lets you undo the function for any output in one step. For example, a weather app that shows both °C and °F needs a formula each way.

Method 1: reverse the operations

Write down what f does to x, step by step. Then undo each step in reverse order (the socks-and-boots rule).

For f(x) = 3x + 2: f multiplies by 3, then adds 2. The inverse subtracts 2, then divides by 3:

f⁻¹(x) = (x − 2)/3

Method 2: swap x and y, then solve

  1. Replace f(x) with y.
  2. Swap x and y (this is the algebra version of swapping coordinates).
  3. Solve for y.
  4. Replace y with f⁻¹(x) (if the inverse is a function).

Both methods always give the same answer. Method 2 is the one that keeps working when functions get complicated.

Method 1: reverse the operationsf: x → × 3 → + 2 → yf⁻¹: x → − 2 → ÷ 3 → yf⁻¹(x) = (x − 2)/3Method 2: swap x and y, solve for yy = 3x + 2x = 3y + 2 (swap)x − 2 = 3y(x − 2)/3 = yf⁻¹(x) = (x − 2)/3
Two ways to find the inverse of f(x) = 3x + 2. Both give f⁻¹(x) = (x − 2)/3.
Worked example

Example 1: Swap and solve

Find the inverse of f(x) = −(1/2)x + 4. Then check with a point.

  1. Write y = −(1/2)x + 4.
  2. Swap x and y: x = −(1/2)y + 4.
  3. Subtract 4: x − 4 = −(1/2)y.
  4. Multiply both sides by −2: −2(x − 4) = y, so y = −2x + 8.
  5. So f⁻¹(x) = −2x + 8.
  6. Check: f(2) = −1 + 4 = 3, so (2, 3) is on f. Then f⁻¹(3) = −6 + 8 = 2, so (3, 2) is on f⁻¹. ✓

f⁻¹(x) = −2x + 8.

Worked example

Example 2: A fraction in the rule

Find the inverse of g(x) = (2x − 5)/3.

  1. Method 1: read the order carefully. g multiplies by 2, then subtracts 5, then divides by 3.
  2. Reverse: multiply by 3, add 5, divide by 2. So g⁻¹(x) = (3x + 5)/2.
  3. Method 2 to confirm: x = (2y − 5)/3 → 3x = 2y − 5 → 3x + 5 = 2y → y = (3x + 5)/2. ✓
  4. Check: g(4) = (8 − 5)/3 = 1, and g⁻¹(1) = (3 + 5)/2 = 4. ✓

g⁻¹(x) = (3x + 5)/2.

Checking an inverse

If you apply f and then f⁻¹, you must get back to where you started. Algebraically: substitute your answer into f and simplify. You should get exactly x.

For f(x) = 3x + 2 and f⁻¹(x) = (x − 2)/3:

f(f⁻¹(x)) = 3 · (x − 2)/3 + 2 = (x − 2) + 2 = x ✓

Patterns between a line and its inverse

Graph several lines with their inverses and you will notice:

  • If f has slope m (m ≠ 0), then f⁻¹ has slope 1/m. Swapping x and y swaps "rise" and "run".
  • The y-intercept of f becomes the x-intercept of f⁻¹, and vice versa.
  • The lines meet on y = x (unless they are parallel to it, or the same line).
  • A non-horizontal line has domain {x ∈ ℝ} and range {y ∈ ℝ}, and so does its inverse.
  • A horizontal line y = k (slope 0) has the vertical line x = k as its inverse: not a function.
xy134567−3−2134567(0, 2)(2, 0)(−1, −1)1331f(x) = 3x + 2f⁻¹(x) = (x − 2)/3y = xslope of f: 3slope of f⁻¹: 1/3(reciprocals)y-intercept of f: 2x-intercept of f⁻¹: 2(coordinates swap)
f(x) = 3x + 2 rises 3 for every 1 across; f⁻¹ rises 1 for every 3 across. The intercepts (0, 2) and (2, 0) swap, and the lines meet at (−1, −1) on y = x.
Worked example

Example 3: Investigating patterns

Find the inverse of each function and describe what you notice. (a) f(x) = x (b) g(x) = −x + 6 (c) h(x) = 4x

  1. (a) Swap: x = y, so f⁻¹(x) = x. The line y = x is its own reflection.
  2. (b) Swap: x = −y + 6, so y = −x + 6. g⁻¹(x) = −x + 6: g is its own inverse! The line has slope −1, so it is perpendicular to y = x, and reflecting it in y = x lands it on itself.
  3. (c) Swap: x = 4y, so h⁻¹(x) = x/4 (or (1/4)x). Slope 4 becomes slope 1/4.
  4. Pattern: a line is its own inverse when it is y = x or has slope −1.

f⁻¹(x) = x; g⁻¹(x) = −x + 6; h⁻¹(x) = (1/4)x.

Worked example

Example 4: Inverses in context

A ride-share charges $3.50 plus $1.25 per kilometre: C(d) = 1.25d + 3.50, for trips up to 40 km.

(a) Find the inverse and explain what it calculates. (b) A trip cost $22.25. How far was it? (c) State the domain and range of C and of its inverse.

  1. (a) y = 1.25d + 3.50. Swap and solve: d = (C − 3.50)/1.25. In context we keep the letters: the inverse d(C) = (C − 3.50)/1.25 takes a cost and gives the distance.
  2. (b) d(22.25) = (22.25 − 3.50)/1.25 = 18.75/1.25 = 15 km.
  3. (c) C: domain {d ∈ ℝ | 0 ≤ d ≤ 40}. C(0) = 3.50 and C(40) = 53.50, so range {C ∈ ℝ | 3.50 ≤ C ≤ 53.50}.
  4. The inverse swaps them: domain {C ∈ ℝ | 3.50 ≤ C ≤ 53.50}, range {d ∈ ℝ | 0 ≤ d ≤ 40}.

d(C) = (C − 3.50)/1.25; a $22.25 trip is 15 km; the domain and range swap.

Keep the meaning in context

In a real situation, swapping the letters x and y can make the variables lose their meaning (C would suddenly mean distance). It is clearer to solve for the other variable and keep each letter's meaning, as in Example 4.

Just for fun

A function and its inverse walk into a café. The function orders a double-double. The inverse immediately un-orders it. The barista needs a break.

Hands-on activity

Inverse pairs on a calculator

You need: a calculator, paper and a pencil, and a partner if possible.

  1. Your partner chooses a secret linear function, such as f(x) = 5x − 3, and tells you only its rule.
  2. Find f⁻¹ by Method 2 and write it down.
  3. Your partner picks any number, puts it into f, and tells you only the output.
  4. Put that output into your f⁻¹. Did you get the partner's number?
  5. Repeat with f(x) = (x + 4)/2 and f(x) = −3x + 1.

What to look for: f⁻¹(x) = (x + 3)/5, then f⁻¹(x) = 2x − 4, then f⁻¹(x) = (1 − x)/3 (also written −(x − 1)/3). For example, with f(x) = 5x − 3 and a secret number 7: f(7) = 32 and (32 + 3)/5 = 7. Every time, the inverse recovers the secret number.

Your turn

1

Find the inverse of f(x) = x + 9.

2

Find the inverse of f(x) = 5x − 1.

3

Find the inverse of g(x) = (x − 4)/2.

4

Find the inverse of h(x) = −3x + 6.

5

Check that f(x) = 2x − 7 and g(x) = (x + 7)/2 are inverses.

6

A line has slope 5. What is the slope of its inverse?

7

Does f(x) = 8 have an inverse that is a function? Explain.

More practice: warm-up

1

Inverse of f(x) = 6x?

2

Inverse of f(x) = x − 12?

3

What is the first step of Method 2?

4

If f(x) has y-intercept 7, what is the x-intercept of f⁻¹?

5

Is f(x) = −x its own inverse?

More practice: on your own

1

Find the inverse of f(x) = (3/4)x + 6.

2

Find the inverse of f(x) = (5 − 2x)/7.

3

f(x) = 4x − 3. Find f⁻¹(9) two ways: with the inverse formula, and by solving f(x) = 9.

4

Convert Fahrenheit to Celsius by finding the inverse of F = 1.8C + 32. Then convert −40°F.

5

Where do f(x) = 2x − 3 and its inverse intersect?

Challenge

1

For which value of b is f(x) = −x + b its own inverse? For which value of m is f(x) = mx + 4 its own inverse?

2

f(x) = ax + b has inverse f⁻¹(x) = 2x − 6. Find a and b.

3

A phone plan costs C(m) = 0.15m + 25 dollars for m minutes in a month (0 ≤ m ≤ 1000). Find the inverse, state its domain and range, and explain why the inverse needs a restricted domain in context.

Check your understanding

Quiz

1.

What is the inverse of f(x) = 2x + 10?

2.

In Method 2, what do you do right after writing y = f(x)?

3.

A linear function has slope −4. What is the slope of its inverse?

4.

Which function is its own inverse?

5.

If f(x) = (x + 1)/4, what is f⁻¹(5)?

6.

Which statement about f(x) = 3x + 2 and f⁻¹(x) = (x − 2)/3 is true?

For parents and teachers

Big idea: Expectation A1.7 asks students to find the equation of the inverse of a linear function (and, next lesson, a quadratic one), to use function notation, and to notice that "the inverse of a linear function involves applying the inverse operations in the reverse order". A1.4 and A1.6 continue from the previous lesson.

How to help: Give your teen a linear function, ask for its inverse, and then ask them to prove it works by choosing a number, running it through f, and then through f⁻¹.

Common difficulties:

  • Reversing operations in the wrong order (dividing before subtracting).
  • Sign errors when dividing by a negative.
  • Writing 1/f(x) instead of the inverse.
  • In context problems, swapping the letters and losing track of what each variable means. Solving for the other variable keeps the meaning.