Inverse Functions

Part 1. Review of Exponential and Logarithm Functions

Enter the functions f and g.

f[x_] :=  Exp[x];
g[x_] :=  Log[x];

Vary x to get values of the composite functions.

x  =  1;
f[g[x]]
g[f[x]]

Plot the composite functions.

Plot[g[f[x]], {x, -3, 3}];
Plot[f[g[x]], {x, -3, 3}];

Answers to the rest of the questions in Part 1:

Part 2. Inverting the Sine Function

Enter the function F.

F[x_] :=  Integrate[1/Sqrt[1 - t^2], {t, 0, x}];

Calculate values of F, and record your results.

Evaluate[F[0.3]]

Vary x to calculate values of the composite functions.

x  = 0.5;
Evaluate[F[Sin[x]]]
Sin[Evaluate[F[x]]]

Plot the composite functions.

Plot[
        Evaluate[F[Sin[x]]], {x, -3, 3}];
Plot[Evaluate[Sin[F[x]]], {x, -3, 3}];

Answer to question 6:

Part 3. Derivatives of Inverse Trig Functions

The steps in Part 3 don't need Mathematica, but you are welcome to enter your own commands if you wish to use it.  Enter your answers below.

Answers:

Part 4. Inverting the Tangent Function

Enter and graph the functions G and Tan[x].

G[x_] :=  
        Integrate[1/(1 + t^2), {t, 0,  x}];
plot1  =  
        Plot[G[x],  {x, -6, 6},  PlotRange -> {-6, 6},  
          DisplayFunction -> Identity];
plot2  =  
        Plot[Tan[x], {x, -6, 6},  PlotRange -> {-6, 6},  
          PlotStyle -> RGBColor[0, 0, 1],  
          DisplayFunction -> Identity];
Show[plot1, plot2,  
        DisplayFunction -> $DisplayFunction];

Graph the composite functions.

Plot[
        Evaluate[G[Tan[x]]], {x, -3, 3}];
Plot[Evaluate[Tan[G[x]]], {x, -3, 3}];

Part 5. Summary

Answers to the summary questions: