PreCalculus Honors Unit Test #1

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No Calculator Allowed

 1 Describe a sequence of transformations that will transform the graph of $y={x}^{3}$ into the graph of $y={\left(x-2\right)}^{3}+3$. (4)

Shift two units right and three units up.

 2 Describe a sequence of transformations that will transform the graph of $y=\frac{1}{x}$ into the graph of $y=\frac{2}{x+3}$. (4)

Multiply the y-values by two and shift three units left.

 3 Sketch the graph of $f\left(x\right)={\left(x-2\right)}^{2}+3$. Label a few key points. (5)

 4 Sketch the graph of $g\left(x\right)=3\sqrt{x-2}$. Label a few key points. (5)

For all questions on this page, let $f\left(x\right)={x}^{2}-4$ and $g\left(x\right)=2x+5$.

 5 Find $\left(f+g\right)\left(x\right)$. (4)

${x}^{2}+2x+1$

 6 Find $\left(\frac{f}{g}\right)\left(x\right)$. (4)

$\frac{{x}^{2}-4}{2x+5}$

 7 Find $\left(f-g\right)\left(3\right)$. (4)

$\left({\left(3\right)}^{2}-4\right)-\left(2\cdot 3+5\right)=-6$

 8 Find $\left(f\cdot g\right)\left(2\right)$. (4)

$\left({2}^{2}-4\right)\left(2\cdot 2+5\right)=0$

 9 Find $\left(f\circ g\right)\left(x\right)$. (4)

${\left(2x+5\right)}^{2}-4=4{x}^{2}+20x+21$

 10 Find $\left(g\circ f\right)\left(x\right)$. (4)

$2\left({x}^{2}-4\right)+5=2{x}^{2}-3$

 11 Find non-trivial functions $f\left(x\right)$ and $g\left(x\right)$ so that $\left(f\circ g\right)\left(x\right)=\sqrt{{x}^{2}+3}$. (3)

How about $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)={x}^{2}+3$?

 12 Are $f\left(x\right)={x}^{3}$ and $g\left(x\right)=\sqrt[5]{x}-2$ inverse functions? Justify your answer. (4)

They are not—when composed, they do not reduce to just x. $g\left(f\left(x\right)\right)=\sqrt[5]{{x}^{3}}-2\ne x$.

 13 Find the inverse function of $p\left(x\right)=\frac{1}{x+2}-3$. (5)

$y=\frac{1}{x+2}-3$ …swap variables first. $x=\frac{1}{y+2}-3$. Now, solve for y: $x+3=\frac{1}{y+2}⇒y+2=\frac{1}{x+3}⇒y=\frac{1}{x+3}-2$. Thus, ${p}^{-1}\left(x\right)=\frac{1}{x+3}-2$.

Page last updated 14:23 2021-02-16