Select your answer (4 out of 9 ) The function of

Questions: Select your answer (4 out of 9 ) The function of f(x)=3x+2 has a domain of -3<x<5. What is the range of f^(-1)(x) ? f(x)>-3 f(x)<3 -3<f(x)<5 -5<f(x)<3 There is no range for the inverse function.

Transcript text: Select your answer (4 out of 9 ) The function of $f(x)=3 x+2$ has a domain of $-3-3$ $\mathrm{f}(\mathrm{x})<3$ $-3

Solution

Solution Steps

Step 1: Finding the Inverse Function

The given function is $ f(x) = 3x + 2 $. To find the inverse function, we solve for $ x $ in terms of $ y $: \[ y = 3x + 2 \implies x = \frac{y}{3} - \frac{2}{3} \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{x}{3} - \frac{2}{3} \]

Step 2: Determining the Domain of the Original Function

The domain of the original function $ f(x) $ is given as $ -3 < x < 5 $. We will now find the corresponding range of the inverse function, which is the same as the domain of $ f(x) $.

Step 3: Calculating the Range of the Inverse Function

To find the range of the inverse function, we evaluate $ f(x) $ at the endpoints of its domain:

For $ x = -3 $: \[ f(-3) = 3(-3) + 2 = -9 + 2 = -7 \]
For $ x = 5 $: \[ f(5) = 3(5) + 2 = 15 + 2 = 17 \] Thus, the range of the inverse function is: \[ -7 < y < 17 \]