Sure, let's find the inverse of the function [tex]\( g(x) = 3x - 5 \)[/tex], and then evaluate the inverse at [tex]\( x = 5 \)[/tex].
### Step-by-Step Solution:
1. Define the Function:
The function given is [tex]\( g(x) = 3x - 5 \)[/tex].
2. Find the Inverse Function:
To find the inverse function [tex]\( g^{-1}(x) \)[/tex], we need to solve the equation [tex]\( y = g(x) \)[/tex] for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
Start with:
[tex]\[ y = 3x - 5 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex]:
[tex]\[ y + 5 = 3x \][/tex]
[tex]\[ x = \frac{y + 5}{3} \][/tex]
So, the inverse function [tex]\( g^{-1}(x) \)[/tex] is:
[tex]\[ g^{-1}(x) = \frac{x + 5}{3} \][/tex]
4. Find [tex]\( g^{-1}(5) \)[/tex]:
Now that we have the inverse function, we can find [tex]\( g^{-1}(5) \)[/tex]:
[tex]\[ g^{-1}(5) = \frac{5 + 5}{3} \][/tex]
[tex]\[ g^{-1}(5) = \frac{10}{3} \][/tex]
Therefore, the value of [tex]\( g^{-1}(5) \)[/tex] is [tex]\( \frac{10}{3} \)[/tex].
