Question

Find all x and y intercepts of the rational function f(x)=2x²-8/x²-2x-3

a) x intercepts 3 and −1,y intercept 8/3
b) x intercepts -2 and 2,y intercept -8/3
c) x intercepts -2 and 2,y intercept 8/3
d) x intercepts -2 and 2 , y intercept 2
e) None of the above a) b) c) d)

Answer 1

The x-intercepts of the function are found by setting the numerator to zero, giving x = -2 and x = 2. The y-intercept is found by evaluating the function at x=0, which results in 8/3. Therefore, the correct answer is c) x-intercepts -2 and 2, y-intercept 8/3.

Step-by-step explanation:

To find the x-intercepts of the function f(x) = \frac{2x^2 - 8}{x^2 - 2x - 3}, we set the numerator equal to zero, because at x-intercepts, the output of the function (y-value) is zero. So, 2x^2 - 8 = 0. Solving the quadratic equation gives us x^2 - 4 = 0, which factored is (x - 2)(x + 2) = 0. This gives us the x-intercepts at x = -2 and x = 2.

For the y-intercept, we find the function value when x=0. Thus, the f(0) = \frac{2 \cdot 0^2 - 8}{0^2 - 2 \cdot 0 - 3} simplifies to f(0) = -\frac{8}{-3} or f(0) = \frac{8}{3}. Therefore, the y-intercept is \frac{8}{3}.

Comparing our solution to the provided options, the correct answer is c) x-intercepts -2 and 2, y-intercept 8/3