Question

find all x-intercepts of the following function. write your answer in answers as coordinate points. be sure to select the appropriate number of x-intercepts.
$f(x) = \frac{x - 5}{x^2 - 7x + 10}$
answer attempt 1 out of x
there is one x-intercept x

Explanation:

Step1: Recall x-intercept definition

To find the \( x \)-intercepts of a function \( f(x) \), we set \( f(x) = 0 \) and solve for \( x \), while ensuring the solution is in the domain of the function (i.e., the denominator is not zero at that \( x \)-value).

Step2: Set \( f(x) = 0 \)

Given \( f(x)=\frac{x - 5}{x^{2}-7x + 10} \), set \( \frac{x - 5}{x^{2}-7x + 10}=0 \). A fraction equals zero when its numerator is zero (and denominator is not zero). So solve \( x - 5 = 0 \).

Step3: Solve numerator equation

From \( x - 5 = 0 \), we get \( x = 5 \).

Step4: Check denominator at \( x = 5 \)

Factor the denominator: \( x^{2}-7x + 10=(x - 2)(x - 5) \). When \( x = 5 \), the denominator is \( (5 - 2)(5 - 5)=3\times0 = 0 \). So \( x = 5 \) is not in the domain of \( f(x) \) (since it makes the denominator zero, the function is undefined there).

Step5: Conclusion on x-intercepts

Since the only solution to \( f(x)=0 \) ( \( x = 5 \)) is not in the domain of the function, there are no \( x \)-intercepts.

Answer:

There is no \( x \)-intercept.