**High School Number Sense Lesson 73: Inverse Functions**

So far we have spent three lessons on functions:

**Function**,

**FunctionCompo**, and

**FunctionExpo**. It may be helpful for you to review these before continuing to today's lesson. Now we will look at the inverse of a function. This concept appeared often last year:

**16 times**with a median placement at

**question # 74**.

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An example of a typical function is:

**f(x) = x + 4**. The inverse of this function would be

**g(y) = y - 4**. An inverse of a function "reverses" the function, so that if the function applied to x gives an output of y, then the inverse function of y gives an output of x (and vice versa). So f(x) and g(y) are inverses of each other if

**f(x) = y and g(y) = x**.

How to Solve:

How to Solve:

- Set y equal to the equation.
- Switch the x and y variables.
- Solve for y.

**Example 1:**

**If f(x) = 2x - 1, then f**

^{-1}(3) = ___1. Change

**y = 2x - 1**to

**x = 2y - 1**

2. Solve for y. 2y = x + 1, so y = (x + 1)/2

3. Plug 3 back in. y = (

**3**+ 1)/2 = 4/2 =

**2**.

**Example 2:**

**If f(x) = 4x + 3, then f**

^{-1}(-2) = ___1. Change

**y = 4x + 3**to

**x = 4y + 3**

2. Solve for y. 4y = x - 3, so y = (x - 3)/4

3. Plug (-2) back in. y = (

**-2**- 3)/4 =

**-5/4**.

**Example 3:**

**If f(x) = 2 + 3/(4 - x), then f**

^{-1}(5) = ___1. Change

**y = 2 + 3/(4 - x)**to

**x = 2 + 3/(4 - y)**

2. Solve for y. x - 2 = 3/(4 - y), so (x - 2)(4 - y) = 3, so 4 - y = 3/(x - 2).

3. 4 = 3/(x - 2) + y, so 4 - 3/(x - 2) = y.

4. Plug 5 back in. y = 4 - 3/(

**5**- 2) = 4 - 3/3 = 4 - 1 =

**3**.

**Example 4:**

**If f(x) = 1 - (2x/3), then f**

^{-1}(4) = ___1. Change

**y = 1 - (2x/3)**to

**x = 1 - (2y/3)**.

2. Solve for y. x + (2y/3) = 1, so 2y/3 = 1 - x.

3. 2y = 3(1 - x), so 2y = 3 - 3x, so y = (3 - 3x)/2.

4. Plug 4 back in. y = [3 - 3(

**4**)]/2 = (3 - 12)/2 =

**-9/2**.

**Example 5:**

**If f(x) = 5x + 3, then f[f**

^{-1}(2)] = ___*

*Note: this combines inverse functions with function compositions.*

1. Change

**y = 5x + 3**to

**x = 5y + 3**.

2. Solve for y. 5y = x - 3, so y = (x - 3)/5.

3. Plug 2 back in. y = (

**2**- 3)/5 =

**-1/5**.

4. Now plug

**-1/5**back into the original function.

5. f(-1/5) = 5(-1/5) + 3 = -1 + 3 =

**2**.

6.

*Notice that this was your original*

**x**value. This should happen regardless of the x chosen, and regardless of the function, as long as it is invertible. Save yourself some steps!**Check back soon for a free worksheet to help you practice FunctionInver.**

**Up Next for High School: FunctionDeriv**