Questions: HW16 Rational Functions: Problem 2 (5 points) Suppose (f(x)=frac2 x^2-4x^3-6). 1. Evaluate (f(5)= (46) /(119)) 2. What are the (x)-intercept of (f(x)) ? Write your answer as an ordered pair. (sqrt(2),0),(-sqrt(2),0) 3. What is the (y)-intercept of (f(x)) ? Write your answer as an ordered pair. ((0,(2) /(3))) 4. Write the equation of the vertical asymptote of (f(x)) : 5. Write the equation of the horizontal asymptote of (f(x)) :

Solution Steps

1. To evaluate \( f(5) \), substitute \( x = 5 \) into the function \( f(x) = \frac{2x^2 - 4}{x^3 - 6} \) and simplify the expression.
2. To find the \( x \)-intercepts, set the numerator of the function equal to zero and solve for \( x \). The \( x \)-intercepts occur where the function equals zero, i.e., where \( 2x^2 - 4 = 0 \).
3. To find the \( y \)-intercept, evaluate the function at \( x = 0 \). This is done by substituting \( x = 0 \) into the function and simplifying.

To evaluate \( f(5) \), we substitute \( x = 5 \) into the function: \[ f(5) = \frac{2(5)^2 - 4}{(5)^3 - 6} = \frac{50 - 4}{125 - 6} = \frac{46}{119} \]

The \( x \)-intercepts occur where the function equals zero, which is when the numerator is zero: \[ 2x^2 - 4 = 0 \implies x^2 = 2 \implies x = \pm \sqrt{2} \] Thus, the \( x \)-intercepts are \( (-\sqrt{2}, 0) \) and \( (\sqrt{2}, 0) \).

To find the \( y \)-intercept, we evaluate the function at \( x = 0 \): \[ f(0) = \frac{2(0)^2 - 4}{(0)^3 - 6} = \frac{-4}{-6} = \frac{2}{3} \] Thus, the \( y \)-intercept is \( (0, \frac{2}{3}) \).

Final Answer