Question

select the correct answer from each drop-down menu.

f(x) = x² - 8x + 15

g(x) = x - 3

h(x) = f(x) ÷ g(x)

h(x) = . the domain of h(x) is ∪ .

Explanation:

Step1: Factorize \( f(x) \)

Factor \( f(x) = x^2 - 8x + 15 \). We need two numbers that multiply to \( 15 \) and add to \( -8 \). Those numbers are \( -3 \) and \( -5 \). So, \( f(x)=(x - 3)(x - 5) \).

Step2: Simplify \( h(x) \)

Given \( h(x)=\frac{f(x)}{g(x)}=\frac{(x - 3)(x - 5)}{x - 3} \). Cancel out the common factor \( x - 3 \) (for \( x
eq3 \)), so \( h(x)=x - 5 \) (with \( x
eq3 \)).

Step3: Determine the domain

The function \( g(x)=x - 3 \) cannot be zero (since division by zero is undefined), so \( x - 3
eq0\Rightarrow x
eq3 \). The domain of \( h(x) \) is all real numbers except \( x = 3 \), which can be written as \( (-\infty, 3)\cup(3, \infty) \).

Answer:

\( h(x)=x - 5 \) (for \( x
eq3 \)); The domain of \( h(x) \) is \( (-\infty, 3)\cup(3, \infty) \)