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C. [tex]\[ \quad f(x) = , \text{ for } x \leq \][/tex]
D. [tex]\[ f^{-1}(x) = x - 15 , \text{ for all } x \][/tex]

Verify that the equation is correct.

[tex]\[
\begin{array}{l}
f\left(f^{-1}(x)\right) = f(T) \\
=4 \\
f^{-1}(f(x)) = f^{-1}(\square) \quad \text{Substitute.} \\
=\square \\
\end{array}
\][/tex]

and

[tex]\[
\begin{aligned}
f^{-1}(f(x)) = f^{-1}(\square) & \quad \text{Substitute.} \\
=\square & \quad \text{Simplify.}
\end{aligned}
\][/tex]

Sagot :

GinnyAnswer
To simplify and verify that the inverse function equation is correct, follow these steps:

Given:

1. [tex]\( f(x) = x + 15 \)[/tex]
2. [tex]\( f^{-1}(x) = x - 15 \)[/tex]

We need to verify that [tex]\( f(f^{-1}(x)) = x \)[/tex] and [tex]\( f^{-1}(f(x)) = x \)[/tex].

### Verifying [tex]\( f(f^{-1}(x)) = x \)[/tex]

Start with [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = x - 15 \][/tex]

Then, apply the function [tex]\( f \)[/tex] to [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f(f^{-1}(x)) = f(x - 15) \][/tex]

Substitute [tex]\( f(x) = x + 15 \)[/tex] into the equation:
[tex]\[ f(x - 15) = (x - 15) + 15 \][/tex]

Simplify the equation:
[tex]\[ (x - 15) + 15 = x \][/tex]

Therefore:
[tex]\[ f(f^{-1}(x)) = x \][/tex]

### Verifying [tex]\( f^{-1}(f(x)) = x \)[/tex]

Start with [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x + 15 \][/tex]

Then, apply the inverse function [tex]\( f^{-1} \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(x + 15) \][/tex]

Substitute [tex]\( f^{-1}(x) = x - 15 \)[/tex] into the equation:
[tex]\[ f^{-1}(x + 15) = (x + 15) - 15 \][/tex]

Simplify the equation:
[tex]\[ (x + 15) - 15 = x \][/tex]

Therefore:
[tex]\[ f^{-1}(f(x)) = x \][/tex]

Both verifications show that the function [tex]\( f \)[/tex] and its inverse [tex]\( f^{-1} \)[/tex] satisfy the required conditions:
[tex]\[ f(f^{-1}(x)) = x \][/tex]
[tex]\[ f^{-1}(f(x)) = x \][/tex]

This confirms that the given inverse function [tex]\( f^{-1}(x) = x - 15 \)[/tex] is correct.

Now, let’s solve a related problem as mentioned:

Problem:
There were nine computers in the server room. Five more computers were installed each day, from Monday to Thursday. How many computers are now in the server room?

Solution:

- Initially, there were 9 computers.
- 5 computers were added each day.
- The number of days between Monday and Thursday (inclusive) is 4 days.

Then:
- In 4 days, the total number of computers added is [tex]\( 5 \times 4 = 20 \)[/tex].

Now, we add the initial number of computers to the new computers:
[tex]\[ 9 + 20 = 29 \][/tex]

Therefore:
- The total number of computers in the server room is 29.
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