Question

1. express the given function (9x - 1)^4 as a composition of two functions f and g such that h(x) = f(g(x)).

a. f(x) = x - 1 and g(x) = 9x^4
b. f(x) = 9x^4 and g(x) = x - 1
c. f(x) = x^4 and g(x) = 9x - 1
d. f(x) = 9x - 1 and g(x) = x^4
e. f(x) = x^4 and g(x) = 9x^4

1. solve the following system of equations. express your solutions as ordered pairs.

\

$$\begin{cases}x^2 + 16x + 23 = y\\\\y = x - 3\\end{cases}$$

a. (-13, -16), (-2, -5)
b. (5, 2), (16, 13)
c. (-10, 4), (3, 6)
d. (1, -2), (4, 1)
e. no solution

1. solve the following equation:

\frac{3x}{x - 5} - 2 = \frac{2x - 2}{x - 5}
a. 5
b. 12
c. 17
d. 8
e. no solution

Explanation:

Question 8

Step1: Recall function composition

Function composition \( h(x) = f(g(x)) \) means we substitute \( g(x) \) into \( f \). For \( h(x)=(9x - 1)^4 \), we need to find \( f \) and \( g \) such that \( f(g(x))=(9x - 1)^4 \).

Step2: Analyze each option

• Option A: \( f(g(x))=(9x^4)-1

eq(9x - 1)^4 \)

• Option B: \( f(g(x))=9(x - 1)^4

eq(9x - 1)^4 \)

• Option C: \( f(g(x))=(9x - 1)^4 \), which matches \( h(x) \)
• Option D: \( f(g(x))=9x^4 - 1

eq(9x - 1)^4 \)

• Option E: \( f(g(x))=(9x^4)^4

eq(9x - 1)^4 \)

Step1: Substitute \( y = x - 3 \) into the first equation

Substitute \( y \) in \( x^2 + 16x + 23 = y \) to get \( x^2 + 16x + 23=x - 3 \)

Step2: Rearrange into quadratic equation

\( x^2 + 16x - x+23 + 3 = 0 \) → \( x^2 + 15x + 26 = 0 \)

Step3: Factor the quadratic

\( x^2 + 15x + 26=(x + 13)(x + 2)=0 \)

Step4: Solve for \( x \)

\( x=-13 \) or \( x = - 2 \)

Step5: Find corresponding \( y \)

For \( x=-13 \), \( y=-13 - 3=-16 \); for \( x=-2 \), \( y=-2 - 3=-5 \)

Step1: Multiply both sides by \( x - 5 \) (note \( x

eq5 \))
\( 3x-2(x - 5)=2x - 2 \)

Step2: Simplify left side

\( 3x-2x + 10=2x - 2 \) → \( x + 10=2x - 2 \)

Step3: Solve for \( x \)

\( 10 + 2=2x - x \) → \( x = 12 \)

Step4: Check domain

\( x = 12
eq5 \), so it's a valid solution

Answer:

C. \( f(x) = x^4 \) and \( g(x) = 9x - 1 \)

Question 9