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Let's break down the problem step by step.

We are given the following:

1. **Functions**:
   - \( f(x) = 2x + 5 \)
   - \( g(x) = x^2 - 3 \)
   - Function \( h \) is defined such that:
     - \( f(p) = q \)
     - \( g(q) = r \)
     - \( h(p) = r \)

### (a) If \( q = 7 \), find the values of \( p \) and \( r \).

We know from the problem setup that \( f(p) = q \). So, we can substitute \( q = 7 \) into the equation for \( f(p) \):
\[
f(p) = 2p + 5 = 7
\]
Solve for \( p \):
\[
2p = 7 - 5 = 2 \quad \Rightarrow \quad p = \frac{2}{2} = 1
\]
So, \( p = 1 \).

Next, to find \( r \), we use the equation \( g(q) = r \) and substitute \( q = 7 \) into the equation for \( g(x) \):
\[
g(7) = 7^2 - 3 = 49 - 3 = 46
\]
So, \( r = 46 \).

### Answer for part (a):
- \( p = 1 \)
- \( r = 46 \)

### (b) Find a formula for \( h(x) \) in terms of \( x \).

We know from the problem that \( h(p) = r \) and that \( r = g(q) \) and \( f(p) = q \).

This means that \( h(p) = g(f(p)) \). So, the function \( h(x) \) can be written as the composition of the functions \( f \) and \( g \):
\[
h(x) = g(f(x))
\]
Substitute the expression for \( f(x) \) into \( g \):
\[
h(x) = g(2x + 5)
\]
Now, use the expression for \( g(x) = x^2 - 3 \) and substitute \( 2x + 5 \) for \( x \):
\[
h(x) = (2x + 5)^2 - 3
\]
Expand the square:
\[
h(x) = (4x^2 + 20x + 25) - 3 = 4x^2 + 20x + 22
\]
Thus, the formula for \( h(x) \) is:
\[
h(x) = 4x^2 + 20x + 22
\]

### Final answers:
- (a) \( p = 1 \) and \( r = 46 \)
- (b) \( h(x) = 4x^2 + 20x + 22 \)

Would you like more details or explanations for any part of this solution? Here are some follow-up questions for deeper understanding:

1. How does the composition of functions \( h(x) = g(f(x)) \) work in general?
2. Can you find \( h(1) \) using the formula \( h(x) = 4x^2 + 20x + 22 \)?
3. What is the general process for solving for \( p \) when given \( q \)?
4. Can you verify that \( g(7) = 46 \) by substituting into \( g(x) \)?
5. What are some other examples where composition of functions is useful?

**Tip**: When solving problems involving composition of functions, always substitute one function into another step-by-step to avoid errors.

The diagram illustrates three functions f, g, and h. The functions are defined by f(x) = 2x + 5 and g(x) = x^2 - 3. The function h is such that whenever f(p) = q and g(q) = r, then h(p) = r. (a) If q = 7, find the values of p and r. (b) Find a formula for h(x), in terms of x.

Learn how to solve a function composition problem involving f(x) = 2x + 5 and g(x) = x^2 - 3. Find the values of p and r when q = 7, and derive the formula for h(x) using function composition techniques. Suitable for high school students in Grades 9-12.

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