Question

practice - rewrite each rational function in the simplest terms. include restrictions on the variable.
e. $f(x) = \frac{2x - 6}{4x - 12}$
f. $f(x) = \frac{5x + 40}{4x + 32}$
g. $f(x) = \frac{x^2 + x - 30}{3x^2 + 18x}$
h. $f(x) = \frac{x^2 + 9x + 20}{2x + 8}$
i. $f(x) = \frac{x^2 - 5x + 6}{x^2 + 2x - 15}$
j. $f(x) = \frac{x^2 - 3x - 10}{x^2 + x - 2}$

Explanation:

Part e: \( f(x) = \frac{2x - 6}{4x - 12} \)

Step 1: Factor numerator and denominator

Factor \( 2x - 6 \) as \( 2(x - 3) \), and \( 4x - 12 \) as \( 4(x - 3) \). So the function becomes \( \frac{2(x - 3)}{4(x - 3)} \).

Step 2: Simplify the fraction

Cancel out the common factor \( (x - 3) \) (note \( x
eq 3 \) to avoid division by zero). Then simplify \( \frac{2}{4} \) to \( \frac{1}{2} \).

Step 1: Factor numerator and denominator

Factor \( 5x + 40 \) as \( 5(x + 8) \), and \( 4x + 32 \) as \( 4(x + 8) \). The function is now \( \frac{5(x + 8)}{4(x + 8)} \).

Step 2: Simplify the fraction

Cancel the common factor \( (x + 8) \) (so \( x
eq -8 \)). The simplified fraction is \( \frac{5}{4} \).

Step 1: Factor numerator and denominator

Factor the numerator: \( x^2 + x - 30=(x + 6)(x - 5) \). Factor the denominator: \( 3x^2 + 18x = 3x(x + 6) \). The function becomes \( \frac{(x + 6)(x - 5)}{3x(x + 6)} \).

Step 2: Simplify the fraction

Cancel the common factor \( (x + 6) \) (note \( x
eq -6 \) and \( x
eq 0 \) to avoid division by zero). The simplified form is \( \frac{x - 5}{3x} \).

Answer:

\( f(x) = \frac{1}{2} \), with the restriction \( x
eq 3 \)

Part f: \( f(x) = \frac{5x + 40}{4x + 32} \)