Given f(x)f(x), find g(x)g(x) and h(x)h(x) such that f(x)=g(h(x))f(x)=g(h(x)) and neither g(x)g(x) nor h(x)h(x) is solely x.

f(x)=−4/(3x−2)


To find functions \(g(x)\) and \(h(x)\) such that \(f(x) = g(h(x))\) and neither \(g(x)\) nor \(h(x)\) is solely \(x\), let's consider the given function:

\[
f(x) = -\frac{4}{3x - 2}
\]

### Step-by-Step Solution

We want to express \(f(x)\) as \(g(h(x))\), where neither \(g(x)\) nor \(h(x)\) is just \(x\).

#### Step 1: Choose a form for \(h(x)\)

One approach is to look at the inner expression \(3x - 2\) and consider this as our \(h(x)\). Let’s set:

\[
h(x) = 3x - 2
\]

This choice ensures that \(h(x)\) is not just \(x\), but a linear function.

#### Step 2: Determine \(g(x)\)

Given that \(h(x) = 3x - 2\), we want to find \(g(x)\) such that:

\[
f(x) = g(h(x)) = g(3x - 2)
\]

Substituting \(h(x)\) into \(f(x)\), we get:

\[
f(x) = -\frac{4}{h(x)}
\]

So we can define \(g(x)\) based on this relationship:

\[
g(x) = -\frac{4}{x}
\]

#### Step 3: Verify the decomposition

Now we have:

\[
g(x) = -\frac{4}{x}, \quad h(x) = 3x - 2
\]

Let's verify that \(f(x) = g(h(x))\):

\[
g(h(x)) = g(3x - 2) = -\frac{4}{3x - 2}
\]

Which is exactly \(f(x)\).

### Conclusion

Thus, one possible pair of functions \(g(x)\) and \(h(x)\) such that \(f(x) = g(h(x))\) are:

\[
\boxed{g(x) = -\frac{4}{x}, \quad h(x) = 3x - 2}
\]

Neither \(g(x)\) nor \(h(x)\) is solely \(x\).

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. Can you find another pair of functions \(g(x)\) and \(h(x)\) that would work for the same \(f(x)\)?
2. How would the decomposition change if \(f(x)\) had a more complicated numerator or denominator?
3. What are some examples of decomposing quadratic functions into two non-quadratic functions?
4. How does function composition affect the domain and range of the resulting function?
5. What strategies can be used to decompose a rational function like \(f(x) = \frac{ax + b}{cx + d}\)?

**Tip:** When decomposing a function into two others, try to identify a part of the expression that could be isolated as one function, making the rest easier to express as a composition.

Explore how to decompose the rational function f(x) = -4/(3x - 2) into two non-linear functions g(x) and h(x), satisfying the conditions where neither g(x) nor h(x) is solely x. This detailed solution covers function composition and rational functions.

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