Question

find the coordinates of the x-intercepts for the following function, if they exist
f(x)=x² - 4x + 3
find any x-intercepts of the graph. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x-intercept(s) is(are) .
(simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. there is no x-intercept.

Explanation:

Step1: Set \(f(x) = 0\)

To find the x - intercepts, we set \(y = f(x)=0\). So we have the equation \(x^{2}-4x + 3=0\).

Step2: Factor the quadratic equation

We factor the quadratic expression \(x^{2}-4x + 3\). We need two numbers that multiply to \(3\) and add up to \(- 4\). The numbers are \(-1\) and \(-3\). So, \(x^{2}-4x + 3=(x - 1)(x - 3)\).
The equation becomes \((x - 1)(x - 3)=0\).

Step3: Solve for \(x\)

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So, if \((x - 1)(x - 3)=0\), then \(x-1 = 0\) or \(x - 3=0\).

• If \(x-1 = 0\), then \(x = 1\).
• If \(x - 3=0\), then \(x = 3\).

For a function \(y = f(x)\), the x - intercepts are the points \((x,0)\) where \(x\) is the solution of \(f(x)=0\). So the x - intercepts are \((1,0)\) and \((3,0)\).

Answer:

A. The x - intercept(s) is(are) \((1,0),(3,0)\)