91Ó°ÊÓ

$$\text {Solve the following equations.}$$ $$5^{3 x}=29$$

Use the table to evaluate the given compositions. $$\begin{array}{lrrrrrr}\hline x & -1 & 0 & 1 & 2 & 3 & 4 \\\f(x) & 3 & 1 & 0 & -1 & -3 & -1 \\\g(x) & -1 & 0 & 2 & 3 & 4 & 5 \\ h(x) & 0 & -1 & 0 & 3 & 0 & 4 \\\\\hline\end{array}$$ a. \(h(g(0))\) b. \(g(f(4))\) c. \(h(h(0))\) d. \(g(h(f(4)))\) e. \(f(f(f(1)))\) f. \(h(h(h(0)))\) g. \(f(h(g(2)))\) h. \(g(f(h(4)))\) i. \(g(g(g(1)))\) j. \(f(f(h(3)))\)

Using inverse relations One hundred grams of a particular radioactive substance decays according to the function \(m(t)=100 e^{-t / 650},\) where \(t>0\) measures time in years. When does the mass reach 50 grams?

Simplify the difference quotient\(\frac{f(x+h)-f(x)}{h}\) for the following functions. $$f(x)=x^{2}$$

Use analytical methods to find the following points of intersection. Find the point(s) of intersection of the parabolas \(y=x^{2}\) and \(y=-x^{2}+8 x\).

Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\sin ^{-1} x\right)$$

Simplify the difference quotient\(\frac{f(x+h)-f(x)}{h}\) for the following functions. $$f(x)=4 x-3$$

Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\sin ^{-1}(x / 3)\right)$$

The population \(P\) of a small town grows according to the function \(P(t)=100 e^{t / 50},\) where \(t\) measures the number of years after \(2010 .\) How long does it take the population to double?

Find a simple function that fits the data in the tables. $$\begin{array}{|r|r|} \hline x & y \\ \hline-1 & 0 \\ \hline 0 & 1 \\ \hline 1 & 2 \\ \hline 2 & 3 \\ \hline 3 & 4 \\ \hline \end{array}$$