Question

Number 14. Directions in pic. And also when you graph do the main function in red and the inverse in blue

Answer 1

Question 14.

Given the function:

`f(x)=-(2)/(3)x-4`

Let's find the inverse of the function.

To find the inverse, take the following steps.

Step 1.

Rewrite f(x) for y

`y=-(2)/(3)x-4`

Step 2.

Interchange the variables:

`x=-(2)/(3)y-4`

Step 3.

Solve for y

Add 4 to both sides:

`\begin{gathered} x+4=-(2)/(3)y-4+4 \\ \\ x+4=-(2)/(3)y \end{gathered}`

Multply all terms by 3:

`\begin{gathered} 3x+3(4)=-(2)/(3)y\ast3 \\ \\ 3x+12=-2y \end{gathered}`

Divide all terms by -2:

`\begin{gathered} -(3)/(2)x+(12)/(-2)=(-2y)/(-2) \\ \\ -(3)/(2)x-6=y \\ \\ y=-(3)/(2)x-6 \end{gathered}`

Therefore, the inverse of the function is:

`f^(-1)(x)=-(3)/(2)x-6`

Let's graph both functions.

To graph each function let's use two points for each.

• Main function:

Find two point usnig the function.

When x = 3:

`\begin{gathered} f(3)=-(2)/(3)\ast3-4 \\ \\ f(3)=-2-4 \\ \\ f(3)=-6 \end{gathered}`

When x = 0:

`\begin{gathered} f(0)=-(2)/(3)\ast(0)-4 \\ \\ f(-3)=-4 \end{gathered}`

For the main function, we have the points:

(3, -6) and (0, -4)

Inverse function:

When x = 2:

`\begin{gathered} f^(-1)(2)=-(3)/(2)\ast(2)-6 \\ \\ f^(-1)(2)=-3-6 \\ \\ f^1(2)=-9 \end{gathered}`

When x = -2:

`\begin{gathered} f^(-1)(-2)=-(3)/(2)\ast(-2)-6 \\ \\ f^1(-2)=3-6 \\ \\ f^(-1)(2)=-3 \end{gathered}`

For the inverse function, we have the points:

(2, -9) and (-2, -3)

To graph both functions, we have:

ANSWER:

`\begin{gathered} \text{ Inverse function:} \\ f^(-1)(x)=-(3)/(2)x-6 \end{gathered}`