Question

question 4 (5 points)
for the functions ( f(x) = 3x - 8 ) and ( g(x) = 2x + 3 ). find each expression. simplify completely

a. ( 2f(x) + g(x) )

b. ( 3g(x) - f(x) )

c. ( f(x) cdot g(x) )

d. ( \frac{f(6)}{g(6)} )

e. ( f(g(x)) )

Explanation:

Part a: \( 2f(x) + g(x) \)

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

We know \( f(x) = 3x - 8 \) and \( g(x) = 2x + 3 \). Substitute these into the expression:
\( 2(3x - 8) + (2x + 3) \)

Step 2: Distribute the 2

Using the distributive property \( a(b + c) = ab + ac \), we get:
\( 6x - 16 + 2x + 3 \)

Step 3: Combine like terms

Combine the \( x \)-terms and the constant terms:
\( (6x + 2x) + (-16 + 3) = 8x - 13 \)

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

Substitute \( f(x) = 3x - 8 \) and \( g(x) = 2x + 3 \) into the expression:
\( 3(2x + 3) - (3x - 8) \)

Step 2: Distribute the 3 and the negative sign

Distribute the 3 to \( 2x + 3 \) and the negative sign to \( 3x - 8 \):
\( 6x + 9 - 3x + 8 \)

Step 3: Combine like terms

Combine the \( x \)-terms and the constant terms:
\( (6x - 3x) + (9 + 8) = 3x + 17 \)

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

Substitute \( f(x) = 3x - 8 \) and \( g(x) = 2x + 3 \) into the product:
\( (3x - 8)(2x + 3) \)

Step 2: Use the distributive property (FOIL method)

Multiply each term in the first binomial by each term in the second binomial:
\( 3x \cdot 2x + 3x \cdot 3 - 8 \cdot 2x - 8 \cdot 3 \)

Step 3: Simplify each term

Simplify the products:
\( 6x^2 + 9x - 16x - 24 \)

Step 4: Combine like terms

Combine the \( x \)-terms:
\( 6x^2 - 7x - 24 \)

Answer:

\( 8x - 13 \)

Part b: \( 3g(x) - f(x) \)