. Diketahui f(x) = x + 5, g(x) = 4x - 3, dan h(x) = 2x + 1. Pernya

A. (f^ -1 g^ -1 h^ -1 ) (x) = (x + 45)/8

B. (f^ -1 h^ -1 g^ -1 ) (x) = (x - 41)/8

C. (g^ -1 f^ -1 h^ -1 ) (x) = (x - 5)/8

D. (g¯¹¹¹)(x) = x-12 8

E. (h¯¹g¯¹ of¹)(x) = x -1 6 8

To solve this problem, we need to break it down step by step by focusing on the operations related to the composition of inverse functions.

We are given the following functions:

- \( f(x) = x + 5 \)
- \( g(x) = 4x - 3 \)
- \( h(x) = 2x + 1 \)

### Step 1: Find the inverse of each function

To find the inverse of a function, we reverse the operations in the original function. This means solving for \( x \) in terms of \( y \) when \( y = f(x), g(x), h(x) \).

#### Inverse of \( f(x) = x + 5 \):
Let \( y = f(x) \). This gives us:
\[
y = x + 5
\]
Solving for \( x \):
\[
x = y - 5
\]
Thus, \( f^{-1}(x) = x - 5 \).

#### Inverse of \( g(x) = 4x - 3 \):
Let \( y = g(x) \). This gives us:
\[
y = 4x - 3
\]
Solving for \( x \):
\[
x = \frac{y + 3}{4}
\]
Thus, \( g^{-1}(x) = \frac{x + 3}{4} \).

#### Inverse of \( h(x) = 2x + 1 \):
Let \( y = h(x) \). This gives us:
\[
y = 2x + 1
\]
Solving for \( x \):
\[
x = \frac{y - 1}{2}
\]
Thus, \( h^{-1}(x) = \frac{x - 1}{2} \).

---

### Step 2: Verify the given options

Now we will check the composition of inverse functions as described in the options.

#### A. \( (f^{-1} \circ g^{-1} \circ h^{-1})(x) = \frac{x + 45}{8} \)
We need to compute \( f^{-1}(g^{-1}(h^{-1}(x))) \):

1. Start with \( h^{-1}(x) = \frac{x - 1}{2} \).
2. Next, apply \( g^{-1} \) to this:
   \[
   g^{-1}(h^{-1}(x)) = g^{-1}\left( \frac{x - 1}{2} \right) = \frac{\frac{x - 1}{2} + 3}{4} = \frac{x + 5}{8}
   \]
3. Finally, apply \( f^{-1} \) to the result:
   \[
   f^{-1}\left( \frac{x + 5}{8} \right) = \frac{x + 5}{8} - 5 = \frac{x + 5 - 40}{8} = \frac{x - 35}{8}
   \]
Since this doesn't match the given expression \( \frac{x + 45}{8} \), **Option A is incorrect**.

#### B. \( (f^{-1} \circ h^{-1} \circ g^{-1})(x) = \frac{x - 41}{8} \)
We need to compute \( f^{-1}(h^{-1}(g^{-1}(x))) \):

1. Start with \( g^{-1}(x) = \frac{x + 3}{4} \).
2. Next, apply \( h^{-1} \) to this:
   \[
   h^{-1}(g^{-1}(x)) = h^{-1}\left( \frac{x + 3}{4} \right) = \frac{\frac{x + 3}{4} - 1}{2} = \frac{x - 1}{8}
   \]
3. Finally, apply \( f^{-1} \) to the result:
   \[
   f^{-1}\left( \frac{x - 1}{8} \right) = \frac{x - 1}{8} - 5 = \frac{x - 1 - 40}{8} = \frac{x - 41}{8}
   \]
This matches the given expression, so **Option B is correct**.

#### C. \( (g^{-1} \circ f^{-1} \circ h

Diketahui f(x) = x + 5, g(x) = 4x - 3, dan h(x) = 2x + 1. Tentukan komposisi invers dari fungsi-fungsi tersebut, dan pilih jawaban yang benar dari pilihan berikut: A. (f^ -1 g^ -1 h^ -1 ) (x) = (x + 45)/8, B. (f^ -1 h^ -1 g^ -1 ) (x) = (x - 41)/8, C. (g^ -1 f^ -1 h^ -1 ) (x) = (x - 5)/8, D. (g¯¹¹¹)(x) = x-12 8, E. (h¯¹g¯¹ of¹)(x) = x -1 6 8

Solve the composition of inverse functions involving f(x) = x + 5, g(x) = 4x - 3, and h(x) = 2x + 1. Learn how to calculate f^{-1}, g^{-1}, and h^{-1}, and find the correct answer from the given options. Suitable for high school students in Grades 10-12.

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