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Chapter 1: Problem 38

(a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. $$ f(x)=\frac{2 x^{2}-9}{3-x} $$

Short Answer

Expert verified
The zeros of the function \(f(x) =\frac{2 x^{2}-9}{3-x}\) are \(x = - \sqrt{\frac{9}{2}}\) and \(x = \sqrt{\frac{9}{2}}\).

Step by step solution

01

Graph the Function

Using a graphing utility, plot the function \(f(x) =\frac{2 x^{2}-9}{3-x}\). Identify the x-values where the function intersects the x-axis. These are the zeros of the function.
02

Determine the Zero Values

Inspecting the graph, one will find that it intersects the x-axis at two points. These points are the zeros of the function. We are not given specific points on the graph so symbolic representation of zeros will be specified in Step 3.
03

Verify the Zeros Algebraically

To verify the zeros algebraically, set the equation \(f(x) =\frac{2 x^{2}-9}{3-x}=0\). Solve the equation for x. Zeros will be the roots of the equation. Multiply both sides by \(3-x\) to get rid of the denominator. The equation becomes \(2x^{2} - 9 = 0\). Solve for x to get x = \(卤 \sqrt{\frac{9}{2}}\). However, the value \(x = 3\) should be discarded since it would result in undefined function due to the division by zero in the original function.

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