https://file.aiquickdraw.com/mathbot/file/17293924052650mcbtlmh.jpg

Let's break down and solve each of the questions shown in the image one by one.

### (a) Express \( 4x^2 - 12x + 13 \) in the form \( (2x + a)^2 + b \), where \( a \) and \( b \) are constants.

We start by completing the square on the quadratic expression \( 4x^2 - 12x + 13 \).

1. Factor out 4 from the quadratic terms:
   \[
   4(x^2 - 3x) + 13
   \]
2. Complete the square on \( x^2 - 3x \). The coefficient of \( x \) is -3, and half of it squared is \( \left(\frac{-3}{2}\right)^2 = \frac{9}{4} \).
   \[
   4\left( (x - \frac{3}{2})^2 - \frac{9}{4} \right) + 13
   \]
3. Distribute the 4:
   \[
   4(x - \frac{3}{2})^2 - 9 + 13
   \]
   Simplifying:
   \[
   4(x - \frac{3}{2})^2 + 4
   \]
4. Now, rewrite the expression as:
   \[
   (2x - 3)^2 + 4
   \]
   So, the constants \( a \) and \( b \) are:
   \[
   a = -3, \quad b = 4
   \]

### (b) Given that it is possible to form the composite function \( gf \), find the least possible value of \( p \) and the greatest possible value of \( q \).

The function \( f(x) = 4x^2 - 12x + 13 \) is defined for \( p < x < q \), and the function \( g(x) = 3x + 1 \) is defined for \( x < 8 \).

For the composite function \( gf(x) \) to exist, the output of \( f(x) \) (i.e., \( f(x) = 4x^2 - 12x + 13 \)) must be within the domain of \( g(x) \), which is \( (-\infty, 8) \).

We need to solve:
\[
4x^2 - 12x + 13 < 8
\]
Simplify:
\[
4x^2 - 12x + 5 < 0
\]
Divide by 4:
\[
x^2 - 3x + \frac{5}{4} < 0
\]
Find the roots of the corresponding quadratic equation:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(\frac{5}{4})}}{2(1)} = \frac{3 \pm \sqrt{9 - 5}}{2} = \frac{3 \pm \sqrt{4}}{2} = \frac{3 \pm 2}{2}
\]
The solutions are:
\[
x = \frac{3 + 2}{2} = 2.5 \quad \text{and} \quad x = \frac{3 - 2}{2} = 0.5
\]
Therefore, \( f(x) \) is less than 8 between \( x = 0.5 \) and \( x = 2.5 \). So, the least possible value of \( p \) is 0.5, and the greatest possible value of \( q \) is 2.5.

### (c) Find an expression for \( gf(x) \).

We already know:
\[
f(x) = 4x^2 - 12x + 13 \quad \text{and} \quad g(x) = 3x + 1
\]
The composite function \( gf(x) \) is:
\[
gf(x) = g(f(x)) = g(4x^2 - 12x + 13) = 3(4x^2 - 12x + 13) + 1
\]
Simplify:
\[
gf(x) = 12x^2 - 36x + 39 + 1 = 12x^2 - 36x + 40
\]

### (d) Find an expression for \( h^{-1}(x) \), where \( h(x) = 4x^2 - 12x + 13 \) for \( x < 0 \).

We are tasked with finding the inverse function of \( h(x) \). First, express \( h(x) \) as a completed square:
\[
h(x) = (2x - 3)^2 + 4
\]
Let \( y = h(x) \), so:
\[
y = (2x - 3)^2 + 4
\]
Solve for \( x \):
\[
y - 4 = (2x - 3)^2
\]
Take the square root of both sides:
\[
\sqrt{y - 4} = 2x - 3
\]
Since \( x < 0 \), we take the negative square root:
\[
-\sqrt{y - 4} = 2x - 3
\]
Solve for \( x \):
\[
x = \frac{-\sqrt{y - 4} + 3}{2}
\]
Thus, the inverse function is:
\[
h^{-1}(x) = \frac{-\sqrt{x - 4} + 3}{2}
\]

---

Would you like more details on any part of the solution or explanations? Here are five related questions you might consider:

1. What are some other methods for completing the square?
2. How does the domain of a composite function depend on the original functions?
3. Could we solve \( gf(x) \) by applying a different function first?
4. How can we confirm if the calculated inverse function is correct?
5. What happens to the range of \( f(x) \) and \( g(x) \) if their domains are extended?

**Tip:** Always check the domain restrictions when forming composite functions to avoid invalid outputs.

Solve the following problems involving functions and quadratics, including expressing 4x^2 - 12x + 13 in the form (2x + a)^2 + b, finding the least possible value of p and the greatest possible value of q, and finding the inverse of h(x) = 4x^2 - 12x + 13.

Learn to solve quadratic functions by completing the square and explore composite functions with domain and range. Find the inverse of a quadratic function h(x). Detailed solutions for high school students in Grades 10-12.

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