PRAXIS: Exponential Functions, Logarithms, and Building New Functions From Old

Homework protip! Assignments go a lot faster if you work without distractions of any kind! On the flip side of that, if you work WITH distractions, such as checking your phone, it will take you up to 4 times longer to get your work done… and the assignment is far more painful because you’re constantly starting and restarting. See if you can just “embrace the suck” for 30-45 minutes without distraction. I’ll bet you’d get this done well before those 30-45 minutes are up.

1. Let \(f(x)=5^x\). Find \(f(-1)\), \(f(0)\), and \(f(1)\).
2. Graph \(f\) by first plotting the control points of \(f\) and drawing a smooth curve through those points.
3. Let \(g(x)=\left(\frac{3}{4}\right)^x\) . Compute \(g(-2)\) and \(g(3)\).
4. Graph \(g\) (using the same method as #2).

5. Let \(h(x)=\log_3(x)\). Compute \(h(\frac{1}{9})\), \(h(1)\) and \(h(81)\).
6. Graph \(h(x)\) by first plotting the control points of \(h\) and drawing a smooth curve through these points.
7. Let \(J(x)=\log_{\frac{2}{3}}(x)\). Graph \(J\) by first plotting the control points of \(J\).
8. Let \(K(x)=\log_{5}(x)\). Use the change-of-base formula to compute \(K(3)\) and \(K(10)\).

Use the graph below to answer questions 9-12.

9. Compute \((f+g)(-2)\)
10. Compute \((f-g)(0)\)
11. Compute \((f\cdot g)(-1)\)
12. Compute \(\left(\frac{f}{g}\right)(2.5)\)

Let

$$
h(x)=\begin{cases}
x^2-3 & x\leq 0 \\
\frac{1}{x} & 2< x< 3 \\
x^3+2x & 4\leq x
\end{cases}
$$

13. Compute \(h(-1)\), and \(h(2.5)\) if possible
14. Compute \(h(5.5)\) and \(h(3)\) if possible