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Chapter 1: Problem 36

(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. $$ f(x)=\sqrt{3 x-14}-8 $$

Short Answer

Expert verified
The zero of the function \(f(x)=\sqrt{3x-14}-8\) is obtained once graphed and then verified algebraically.

Step by step solution

01

Graphing the Function

To graph the function \(f(x)=\sqrt{3x-14}-8\), plug in the function into a graphing utility. By checking the graph, we can see x-intercepts, or zeros, of the function.
02

Identifying the Zeros

Zeros of a function are the x-values that make the function equal to 0. Looking at the graph, we could identify these values when y=0. Here, the function intersects x-axis at one point, this is the value of x. To calculate the zero of the function, set the function equal to zero and solve for x: \(0=\sqrt{3x-14}-8\). Solving this gives us the zero of the function.
03

Algebraically Verifying the Zeros

From the graphical solution, we have an x-value which is the zero of the function. Now, we need to check this value algebraically. Substitute this x-value into the original function \(f(x)=\sqrt{3x-14}-8\). If this substitution results in the equation being true that means the value is indeed a zero of the function.

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