Consider the functions: 

**f(x)=4x−1f(x)=4x−1**

**h(x)=4x2−1h(x)=4x2−1**


**g(x)=kx2g(x)=kx2** 

**If necessary, give answers as fractions in simplest for. For example: -3/2**


(a) Find the value of ***hf(2)hf(2) = Answer 1 Question 1***

(b) Find the value of ***hf(3/4)hf(3/4) = Answer 2 Question 1***

(c) Find the composite function

explain ncause this is new t9o me

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.

Consider the functions:

f(x) = 4x - 1
h(x) = 4x^2 - 1
g(x) = kx^2

If necessary, give answers as fractions in simplest form.

(a) Find the value of h(f(2))
(b) Find the value of h(f(3/4))
(c) Find the composite function h(f(x))

Learn how to solve composite function problems using linear and quadratic functions. This problem involves evaluating h(f(x)) using the functions f(x) = 4x - 1 and h(x) = 4x^2 - 1. Discover step-by-step solutions for h(f(2)) and h(f(3/4)), and find the general form of h(f(x)). Ideal for students in Grades 9-11.

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