f(x)=3-2x/7 diketahui f(x)=x+4/x+6 dan g(x)=2x-1, rumus fungsi invers (fog)-1(x) adalah

Let's solve for the inverse of the composite function \((f \circ g)^{-1}(x)\) where:

1. \( f(x) = 3 - \frac{2x}{7} \)
2. \( f(x) = \frac{x+4}{x+6} \) (This seems to be another version or mistake. Let's use this later if needed)
3. \( g(x) = 2x - 1 \)

We'll first find \((f \circ g)(x)\), which means substituting \(g(x)\) into \(f(x)\), and then find its inverse.

### Step 1: Find \( (f \circ g)(x) \)
Substitute \(g(x) = 2x - 1\) into \(f(x)\):

\[
(f \circ g)(x) = f(g(x)) = f(2x - 1)
\]

Using \(f(x) = 3 - \frac{2x}{7}\):

\[
f(2x - 1) = 3 - \frac{2(2x - 1)}{7}
\]
\[
= 3 - \frac{4x - 2}{7}
\]
\[
= 3 - \frac{4x}{7} + \frac{2}{7}
\]
\[
= \frac{21}{7} - \frac{4x}{7} + \frac{2}{7}
\]
\[
= \frac{21 + 2 - 4x}{7} 
\]
\[
= \frac{23 - 4x}{7}
\]

So, \((f \circ g)(x) = \frac{23 - 4x}{7}\).

### Step 2: Find the Inverse \( (f \circ g)^{-1}(x) \)

To find the inverse, set \(y = \frac{23 - 4x}{7}\) and solve for \(x\):

\[
y = \frac{23 - 4x}{7}
\]

Multiply both sides by 7:

\[
7y = 23 - 4x
\]

Isolate \(x\):

\[
4x = 23 - 7y
\]

\[
x = \frac{23 - 7y}{4}
\]

Thus, the inverse function \((f \circ g)^{-1}(x)\) is:

\[
(f \circ g)^{-1}(x) = \frac{23 - 7x}{4}
\]

Would you like more details or have any questions about this solution?

Here are some related questions you might find useful:
1. How do you verify if a function and its inverse are correct?
2. What are the domain and range of the inverse function?
3. How do you find the inverse of a non-linear function?
4. Can every function have an inverse? Why or why not?
5. How would the graph of \( (f \circ

Explore how to find the inverse of composite functions (f ∘ g)^{-1}(x) step-by-step. This problem involves composite functions f(x) and g(x), where f(x) = 3 - (2x / 7), f(x) = (x + 4) / (x + 6), and g(x) = 2x - 1. Suitable for undergraduate-level students and anyone interested in advanced algebraic concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation