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Chapter 1: Problem 24

Find the zeros of the function algebraically. $$ f(x)=3 x^{2}+22 x-16 $$

Short Answer

Expert verified
The zeros of the function are \(x = \frac{2}{3}\) and \(x = -8\).

Step by step solution

01

Identify the coefficients a, b and c

In the equation \(f(x) = 3x^2 + 22x - 16\), we have \(a = 3\), \(b = 22\), and \(c = -16\)
02

Substitute the coefficients into the quadratic formula

Substituting \(a = 3\), \(b = 22\), and \(c = -16\) into the quadratic formula gives: \(x = \frac{-22 \pm \sqrt{(22)^2-4*3*(-16)}}{2*3}\)
03

Simplify the expression under the square root

We simplify the expression under the square root before proceeding, this gives: \(x = \frac{-22 \pm \sqrt{484+192}}{6}\)
04

Solve the square root

This simplifies further to: \(x = \frac{-22 \pm \sqrt{676}}{6}\)
05

Rewrite by simplifying the square root

The square root of 676 is 26, substituting and further simplifying gives us: \(x = \frac{-22 \pm 26}{6}\)
06

Find the two roots

Now, we can find the two roots by applying the plus and minus in the formula. This gives us the two roots of the function: \(x = \frac{4}{6}\) or \(x = -8\)

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Most popular questions from this chapter

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{4}{x} $$

Consider the functions given by \(f(x)=x+2\) and \(f^{-1}(x)=x-2 .\) Evaluate \(f\left(f^{-1}(x)\right)\) and \(f^{-1}(f(x))\) for the indicated values of \(x .\) What can you conclude about the functions? $$ \begin{array}{|l|l|l|l|l|} \hline x & -10 & 0 & 7 & 45 \\ \hline f\left(f^{-1}(x)\right) & & & & \\ \hline f^{-1}(f(x)) & & & & \\ \hline \end{array} $$

Find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(y\) varies inversely as \(x .(y=3\) when \(x=25\).

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ (f \circ g)^{-1} $$

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ g^{-1}\right)(-4) $$
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