Question

use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g. 24) g(x)=f(x - 1)-1 multiple choice. choose the one alternative that best completes the statement or answers the question. solve the equation by making an appropriate substitution. 25) x^4 - 40x^2+144 = 0 a) {-2,2,-6,6} b) {2,6} c) {-2i,2i,-6i,6i} d) {4,36} use the vertical line test to determine what

Explanation:

Step1: Analyze transformation for $g(x)$

The function $g(x)=f(x - 1)-1$ is a transformation of $f(x)$. The $x-1$ inside the function shifts $f(x)$ 1 unit to the right, and the $- 1$ outside the function shifts it 1 unit down.

Step2: Solve the equation $x^{4}-40x^{2}+144 = 0$

Let $u = x^{2}$, then the equation becomes $u^{2}-40u + 144=0$.
Using the quadratic formula $u=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $au^{2}+bu + c = 0$ (here $a = 1$, $b=-40$, $c = 144$).
First, calculate the discriminant $\Delta=b^{2}-4ac=(-40)^{2}-4\times1\times144=1600 - 576 = 1024$.
Then $u=\frac{40\pm\sqrt{1024}}{2}=\frac{40\pm32}{2}$.
We get $u_1=\frac{40 + 32}{2}=36$ and $u_2=\frac{40-32}{2}=4$.
Since $u = x^{2}$, when $x^{2}=36$, $x=\pm6$; when $x^{2}=4$, $x=\pm2$.

Answer:

1. Sketch the graph of $y = f(x)$ shifted 1 unit to the right and 1 unit down.
2. A. $\{-2,2,-6,6\}$