Question

find all x-intercepts of the following function. write your answer or answers as coordinate points. be sure to select the appropriate number of x-intercepts.
$f(x) = \frac{x - 7}{3x^2 - 33x + 84}$
answer attempt 1 out of 2
there is one x-intercept

Explanation:

Step1: Recall x - intercept definition

To find the x - intercepts of a function \(y = f(x)\), we set \(y=0\) (i.e., \(f(x) = 0\)) and solve for \(x\). Also, we need to ensure that the \(x\) - value we get does not make the denominator of the rational function equal to zero (since the function is undefined at those points).

For the function \(f(x)=\frac{x - 7}{3x^{2}-23x + 84}\), set \(f(x)=0\). So we have the equation \(\frac{x - 7}{3x^{2}-23x + 84}=0\).

Step2: Solve the numerator for zero

A fraction is zero when its numerator is zero (and the denominator is not zero). So we solve \(x - 7=0\).
Adding 7 to both sides of the equation \(x - 7 = 0\), we get \(x=7\).

Step3: Check the denominator at \(x = 7\)

Now we need to check if \(x = 7\) makes the denominator \(3x^{2}-23x + 84\) equal to zero.
Substitute \(x = 7\) into the denominator: \(3(7)^{2}-23(7)+84=3\times49-161 + 84=147-161 + 84=70
eq0\).
So \(x = 7\) is a valid solution.

Step4: Write the x - intercept as a coordinate point

The x - intercepts are points on the x - axis, so their y - coordinate is 0. So the x - intercept is \((7,0)\).

Answer:

\((7,0)\)