Question

what is the inverse of the function f(x) = 2x - 10?
○ h(x) = 2x - 5
○ h(x) = 2x + 5
○ h(x) = \frac{1}{2}x - 5
○ h(x) = \frac{1}{2}x + 5

Explanation:

To find the inverse of the function \( f(x) = 2x - 10 \), we follow these steps:

Step 1: Replace \( f(x) \) with \( y \)

Start by rewriting the function as:
\( y = 2x - 10 \)

Step 2: Swap \( x \) and \( y \)

To find the inverse, we interchange \( x \) and \( y \) (this reflects the function over the line \( y = x \)):
\( x = 2y - 10 \)

Step 3: Solve for \( y \)

Now, solve the new equation for \( y \):

1. Add 10 to both sides:

\( x + 10 = 2y \)

1. Divide both sides by 2:

\( y = \frac{1}{2}x + 5 \)

Step 4: Replace \( y \) with \( h(x) \) (the inverse function)

The inverse function is:
\( h(x) = \frac{1}{2}x + 5 \)

So the inverse of \( f(x) = 2x - 10 \) is \( \boldsymbol{h(x) = \frac{1}{2}x + 5} \) (the last option).

Answer:

To find the inverse of the function \( f(x) = 2x - 10 \), we follow these steps:

Step 1: Replace \( f(x) \) with \( y \)

Start by rewriting the function as:
\( y = 2x - 10 \)

Step 2: Swap \( x \) and \( y \)

To find the inverse, we interchange \( x \) and \( y \) (this reflects the function over the line \( y = x \)):
\( x = 2y - 10 \)

Step 3: Solve for \( y \)

Now, solve the new equation for \( y \):

1. Add 10 to both sides:

\( x + 10 = 2y \)

1. Divide both sides by 2:

\( y = \frac{1}{2}x + 5 \)

Step 4: Replace \( y \) with \( h(x) \) (the inverse function)

The inverse function is:
\( h(x) = \frac{1}{2}x + 5 \)

So the inverse of \( f(x) = 2x - 10 \) is \( \boldsymbol{h(x) = \frac{1}{2}x + 5} \) (the last option).