If g(x)=2x and f(g(x))=6x-2 then find the value of x such that g(f(x))=10.

To solve for \( x \) such that \( g(f(x)) = 10 \), we need to understand the relationships between the functions \( f \) and \( g \).

### Step 1: Determine \( f(x) \) using the given information

We know:
1. \( g(x) = 2x \)
2. \( f(g(x)) = 6x - 2 \)

Since \( f(g(x)) = 6x - 2 \), let's substitute \( g(x) = 2x \) into \( f \):

\[
f(2x) = 6x - 2
\]

Now, let \( u = 2x \), so \( f(u) = 3u - 2 \) (substituting \( u = 2x \) into \( 6x - 2 \)).

Thus, we have:
\[
f(x) = 3x - 2
\]

### Step 2: Find \( g(f(x)) \)

Now, we want to find \( g(f(x)) \) using \( f(x) = 3x - 2 \):
\[
g(f(x)) = g(3x - 2)
\]

Since \( g(x) = 2x \), we substitute \( 3x - 2 \) into \( g \):
\[
g(3x - 2) = 2(3x - 2) = 6x - 4
\]

### Step 3: Set up the equation \( g(f(x)) = 10 \)

Now, we set \( g(f(x)) = 10 \):
\[
6x - 4 = 10
\]

Solve for \( x \):
\[
6x = 14
\]
\[
x = \frac{14}{6} = \frac{7}{3}
\]

### Final Answer
The value of \( x \) such that \( g(f(x)) = 10 \) is:
\[
x = \frac{7}{3}
\]

Would you like further details on any steps?

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1. What would be the value of \( g(f(x)) \) if \( x = 4 \)?
2. How does \( f(x) \) change if \( f(g(x)) = 8x + 1 \) instead?
3. What happens if we define \( g(x) \) differently, say \( g(x) = 3x \)?
4. How would we solve for \( x \) if \( g(f(x)) = 0 \) instead?
5. How can we find \( x \

Learn to solve for x in the composition of functions problem where g(x) = 2x, f(g(x)) = 6x - 2, and g(f(x)) = 10. This solution involves key algebraic concepts and is ideal for students in Grades 10-12.

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