Question

d. $f(x) = (x - 4)(x + 4)$
$g(x) = x + 4$

Explanation:

Assuming the problem is to find the domain where \( f(x) \) and \( g(x) \) are equal or to simplify \( f(x) \) and analyze its relation to \( g(x) \), here's the step - by - step solution:

Step 1: Simplify \( f(x) \)

We know that \( f(x)=(x - 4)(x + 4) \). By the difference of squares formula \( (a - b)(a + b)=a^{2}-b^{2} \), where \( a = x \) and \( b = 4 \), we get \( f(x)=x^{2}-16 \).

Step 2: Analyze the relation between \( f(x) \) and \( g(x) \)

\( g(x)=x + 4 \). To find when \( f(x)=g(x) \), we set up the equation \( x^{2}-16=x + 4 \).
Rearrange it to the standard quadratic form \( x^{2}-x-20 = 0 \).
Factor the quadratic: We need two numbers that multiply to - 20 and add to - 1. The numbers are - 5 and 4. So, \( (x - 5)(x+4)=0 \).
Set each factor equal to zero:

• For \( x - 5=0 \), we get \( x = 5 \).
• For \( x + 4=0 \), we get \( x=-4 \).

If the problem was just to simplify \( f(x) \), then \( f(x)=x^{2}-16 \). If it was to find the intersection points of \( y = f(x) \) and \( y = g(x) \), the solutions are \( x=-4 \) and \( x = 5 \).

Answer:

If simplifying \( f(x) \): \( f(x)=x^{2}-16 \)
If finding intersection of \( f(x) \) and \( g(x) \): \( x=-4 \) or \( x = 5 \)

(Note: Since the original problem was not fully stated, we provided solutions for common related problems. If you have a specific operation like composition \( f(g(x)) \) or \( g(f(x)) \), here is how it would be done:

For \( f(g(x)) \):

Step 1: Substitute \( g(x) \) into \( f(x) \)

\( f(g(x))=f(x + 4)=((x + 4)-4)((x + 4)+4) \)

Step 2: Simplify

Simplify the terms inside the parentheses: \( (x+4 - 4)=(x) \) and \( (x + 4+4)=(x + 8) \). So \( f(g(x))=x(x + 8)=x^{2}+8x \)

For \( g(f(x)) \):

Step 1: Substitute \( f(x) \) into \( g(x) \)

\( g(f(x))=g(x^{2}-16)=(x^{2}-16)+4 \)

Step 2: Simplify

\( g(f(x))=x^{2}-12 \))