Given  f(x)
 , find  g(x)
  and  h(x)
  such that  f(x)=g(h(x))
  and neither  g(x)
  nor  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 \( x \), we need to decompose the function \( f(x) = -\frac{4}{3}x - 2 \) into a composition of two functions.

### Step-by-Step Solution

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

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

#### Step 1: Identify potential structures for \( h(x) \) and \( g(x) \)

Let's choose a function \( h(x) \) that is neither too simple nor too complex, and then find \( g(x) \) accordingly.

A good strategy is to let \( h(x) \) be a linear function, because the given function \( f(x) \) is also linear. We could choose:

\[
h(x) = ax + b
\]

And then \( g(x) \) will be:

\[
g(x) = c \cdot x + d
\]

#### Step 2: Choose specific forms for \( h(x) \) and \( g(x) \)

Let's try \( h(x) = 2x + 3 \) and find \( g(x) \) such that \( g(h(x)) = f(x) \).

Substitute \( h(x) \) into \( g(x) \):

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

Now, let’s find \( g(x) \).

We need:
\[
g(2x + 3) = -\frac{4}{3}x - 2
\]

To match this, set up a function \( g(x) \):

If \( g(x) = -\frac{2}{3}x + c \), substituting \( x = 2x + 3 \) gives:

\[
g(2x + 3) = -\frac{2}{3}(2x + 3) + c
\]

Simplify:
\[
g(2x + 3) = -\frac{4}{3}x - 2 + c
\]

To match \( f(x) = -\frac{4}{3}x - 2 \), set \( c = 0 \).

So, \( g(x) = -\frac{2}{3}x \).

#### Step 3: Verify the decomposition

Now we have:

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

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

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:

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

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. How can you find a different pair of functions \( g(x) \) and \( h(x) \) for the same \( f(x) \)?
2. What is the composition of two quadratic functions to form a cubic function?
3. Can you decompose a nonlinear function into two linear functions?
4. How does the chain rule in calculus relate to function composition?
5. What are the properties of invertible functions in function decomposition?

**Tip:** When decomposing functions, start by looking for simpler component functions (like linear or quadratic ones) that, when combined, match the original function.

Explore how to decompose the linear function f(x) = -4/3x - 2 into two functions g(x) and h(x), where f(x) = g(h(x)) and neither g(x) nor h(x) is solely x. This problem involves function composition and linear functions, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation