To solve this problem, we need to determine two things given the function [tex]\( f(x) = 2x + 7 \)[/tex]:
1. The expression for [tex]\( f(x+2) \)[/tex]
2. The expression for [tex]\( f^{-1}(x+2) \)[/tex]
Let's start with finding [tex]\( f(x+2) \)[/tex]:
### Step-by-Step Solution for [tex]\( f(x+2) \)[/tex]:
1. Given the original function: [tex]\( f(x) = 2x + 7 \)[/tex]
2. Substitute [tex]\( x+2 \)[/tex] into the function in place of [tex]\( x \)[/tex]:
[tex]\[
f(x+2) = 2(x+2) + 7
\][/tex]
3. Expand and simplify the expression:
[tex]\[
f(x+2) = 2x + 4 + 7
\][/tex]
[tex]\[
f(x+2) = 2x + 11
\][/tex]
So, [tex]\( f(x+2) = 2x + 11 \)[/tex].
### Step-by-Step Solution for [tex]\( f^{-1}(x+2) \)[/tex]:
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we first need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] when [tex]\( y = f(x) \)[/tex].
1. Start with the equation of the original function:
[tex]\[
y = 2x + 7
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
y - 7 = 2x
\][/tex]
[tex]\[
x = \frac{y - 7}{2}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(y) \)[/tex] is:
[tex]\[
f^{-1}(y) = \frac{y - 7}{2}
\][/tex]
Now, substitute [tex]\( x + 2 \)[/tex] for [tex]\( y \)[/tex] in the inverse function:
3. For [tex]\( f^{-1}(x+2) \)[/tex]:
[tex]\[
f^{-1}(x+2) = \frac{(x+2) - 7}{2}
\][/tex]
[tex]\[
f^{-1}(x+2) = \frac{x + 2 - 7}{2}
\][/tex]
[tex]\[
f^{-1}(x+2) = \frac{x - 5}{2}
\][/tex]
So, [tex]\( f^{-1}(x+2) = \frac{x - 5}{2} \)[/tex].
### Summary
- [tex]\( f(x+2) = 2x + 11 \)[/tex]
- [tex]\( f^{-1}(x+2) = \frac{x - 5}{2} \)[/tex]
