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We have come up with the following definitions.
Let’s get a bit of practice using these definitions.
$100^{\left(\frac{1}{2}\right)}$
$100^{-2}$
$100^{\left(\frac{-1}{2}\right)}$
$100^{\left(\frac{3}{2}\right)}$
$100^{\left(\frac{-3}{2}\right)}$
Check all of your answers above on your calculator. If any of them did not come out right, figure out what went wrong, and fix it!
Solve for $x$ : $\frac{{x}^{}}{{x}^{}}17^{\left(\frac{1}{2}\right)}$ $17^{\left(\frac{1}{2}\right)}$
Solve for $x$ : $x^{\left(\frac{1}{2}\right)}=9$
Simplify: $\frac{x}{\sqrt{x}}$
Simplify: $\frac{{x}^{}+\sqrt{x}}{{x}^{}+\frac{1}{\sqrt{x}}}$
Now…remember inverse functions? You find them by switching the $x$ and the $y$ and then solving for $y$ . Find the inverse of each of the following functions. To do this, in some cases, you will have to rewrite the things. For instance, in #9, you will start by writing $y=x^{\left(\frac{1}{2}\right)}$ . Switch the $x$ and the $y$ , and you get $x=y^{\left(\frac{1}{2}\right)}$ . Now what? Well, remember what that means: it means $x=\sqrt{y}$ . Once you’ve done that, you can solve for $y$ , right?
$x^{3}$
$x^{-2}$
$x^{0}$
Can you find a generalization about the inverse function of an exponent?
Graph $y=2^{x}$ by plotting points. Make sure to include both positive and negative $x$ values.
Graph $y=22^{x}$ by doubling all the y-values in the graph of $y=2^{x}$ .
Graph $y=2^{x}+1$ by taking the graph $y=2^{x}$ and “shifting” it to the left by one.
Graph $y=\left(\frac{1}{2}\right)^{x}$ by plotting points. Make sure to include both positive and negative $x$ values.
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