Let’s break down the problem step by step.

(a) Find the value of $h(f(2))$

We need to evaluate the composite function $h(f(2))$, which means we will first find $f(2)$ and then use that result to find $h$ of that value.

Step 1: Find $f(2)$

The function for $f(x)$ is given as: $f(x) = 4x - 1$ Substitute $x = 2$ into $f(x)$: $f(2) = 4(2) - 1 = 8 - 1 = 7$

Step 2: Use this value in $h(x)$

The function for $h(x)$ is given as: $h(x) = 4x^2 - 1$ Now substitute $f(2) = 7$ into $h(x)$: $h(7) = 4(7^2) - 1 = 4(49) - 1 = 196 - 1 = 195$

Thus, $h(f(2)) = 195$.

(b) Find the value of $h(f(3/4))$

Again, we need to find $f(3/4)$ first and then use that result to find $h$.

Step 1: Find $f(3/4)$

Using the same function for $f(x)$: $f(x) = 4x - 1$ Substitute $x = \frac{3}{4}$: $f\left(\frac{3}{4}\right) = 4\left(\frac{3}{4}\right) - 1 = 3 - 1 = 2$

Step 2: Use this value in $h(x)$

Now substitute $f(3/4) = 2$ into $h(x)$: $h(2) = 4(2^2) - 1 = 4(4) - 1 = 16 - 1 = 15$

Thus, $h(f(3/4)) = 15$.

(c) Find the composite function $h(f(x))$

The composite function $h(f(x))$ means we substitute $f(x)$ into $h(x)$. Recall that:

• $f(x) = 4x - 1$
• $h(x) = 4x^2 - 1$

Step 1: Substitute $f(x)$ into $h(x)$

We need to replace every occurrence of $x$ in $h(x)$ with $f(x)$. This means: $h(f(x)) = 4(f(x))^2 - 1$ Substitute $f(x) = 4x - 1$ into this expression: $h(f(x)) = 4(4x - 1)^2 - 1$

Step 2: Expand $(4x - 1)^2$

First, expand the square: $(4x - 1)^2 = (4x - 1)(4x - 1) = 16x^2 - 8x + 1$

Step 3: Multiply by 4 and simplify

Now substitute this back into $h(f(x))$: $h(f(x)) = 4(16x^2 - 8x + 1) - 1 = 64x^2 - 32x + 4 - 1$ Simplify the constant terms: $h(f(x)) = 64x^2 - 32x + 3$

Thus, the composite function $h(f(x)) = 64x^2 - 32x + 3$.

---

Summary of Results:

• $h(f(2)) = 195$
• $h(f(3/4)) = 15$
• $h(f(x)) = 64x^2 - 32x + 3$

Would you like me to go over any part of the explanation in more detail?

---

Here are some related questions to expand your understanding:

1. What happens if you switch the order and find $f(h(x))$?
2. Can you calculate $h(f(-1))$ using the same approach?
3. How would the result change if $f(x) = 2x - 1$ instead of $f(x) = 4x - 1$?
4. What is the general form of $h(g(x))$ if $g(x) = kx^2$?
5. What is the domain of the composite function $h(f(x))$?

Tip:

Always expand composite functions step by step, starting from the innermost function, and substitute carefully.