# Composite functions examples and solutions

Recent questions in Composite functions
Composite functions

### Evaluate the function at given value of the independent variable. Simplify the results. $$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}+{5}}}$$ $$\displaystyle{f{{\left({x}+\triangle{x}\right)}}}$$

Composite functions

### Evaluate the function at given value of the independent variable. Simplify the results. $$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}+{5}}}$$ $$\displaystyle{f{{\left({11}\right)}}}$$

Composite functions

### Use the given graph of f to find a number $$\displaystyle\delta$$ such that if $$\displaystyle{0}{<}{l}{x}-{3}{l}{<}\delta$$ then $$\displaystyle{I}{f{{\left({x}\right)}}}-{2}{I}{<}{0.5}$$

Composite functions

### Explain Composite Functions?

Composite functions

### Finding the Domain of a Composite Function. Find the domain of $$\displaystyle{f}\circ{g}$$ for the functions. $$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}-{9}{\quad\text{and}\quad}{g{{\left({x}\right)}}}=\sqrt{{{9}}}-{x}^{{2}}$$

Composite functions

### For the given functions f and g, find: (a) $$\displaystyle{f}\circ{g}$$ (b) $$\displaystyle{g}\circ{f}$$ (c) $$\displaystyle{f}\circ{f}$$ (d) $$\displaystyle{g}\circ{g}$$ State the domain of each composite function. $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}+{3}}}}.\ {g{{\left({x}\right)}}}=-{\frac{{{2}}}{{{x}}}}$$

Composite functions

### Given $$\displaystyle{f{{\left({x}\right)}}}={4}{x}^{{{2}}}-{3}{\quad\text{and}\quad}{g{{\left({x}\right)}}}={6}-{\frac{{{1}}}{{{2}}}}{x}^{{{2}}}$$ a. (f of g)(4)

Composite functions

### Given $$\displaystyle{f{{\left({x}\right)}}}={4}{x}^{{{2}}}-{3}{\quad\text{and}\quad}{g{{\left({x}\right)}}}={6}-{\frac{{{1}}}{{{2}}}}{x}^{{{2}}}$$ b. $$(g\ of\ f)(2)$$

Composite functions

### In plainspeak, what do composite functions do?

Composite functions

### Given $$\displaystyle{f{{\left({x}\right)}}}={4}{x}^{{{2}}}-{3}{\quad\text{and}\quad}{g{{\left({x}\right)}}}={6}-{\frac{{{1}}}{{{2}}}}{x}^{{{2}}}$$ c. $$(f\ of\ f)(1)$$

Composite functions

### Evaluate the function at given value of the independent variable. Simplify the results. $$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}+{5}}}$$ $$\displaystyle{f{{\left(-{4}\right)}}}$$

Composite functions

### I must have made a mistake in finding the composite functions $$\displaystyle{f}\circ{g}{\quad\text{and}\quad}{g}\circ{f},$$ because I notice that $$\displaystyle{f}\circ{g}$$ is not the same function as $$\displaystyle{g}\circ{f}$$.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

Composite functions

### Evaluate the function at given value of the independent variable. Simplify the results. $$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}+{5}}}$$ $$\displaystyle{f{{\left({4}\right)}}}$$

Composite functions

### Find composite function $$\displaystyle{f}\circ{g}{\quad\text{and}\quad}{g}\circ{f}$$. Find the domain of each composite function. Are the two composite funcctions equal? $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}$$ $$\displaystyle{g{{\left({x}\right)}}}=\sqrt{{{x}}}$$

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{y}={4}^{{-{5}{x}+{2}}}$$

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{y}={3}\dot{{\lbrace}}{4}^{{{x}^{{{2}}}+{2}}}$$

Composite functions

### Find derivatives of the functions defined as follows. $$\displaystyle{f{{\left({z}\right)}}}={\left({2}{z}+{e}^{{-{z}^{{{2}}}}}\right)}^{{{2}}}$$

Composite functions

### Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as $$\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}.{f{{\left({x},{y}\right)}}}={1}-{\frac{{{\cos{{\left({x}^{{{2}}}+{y}^{{{2}}}\right)}}}}}{{{x}^{{{2}}}+{y}^{{{2}}}}}}$$

Composite functions

### Write an integral that requires three applications of integration by parts. Explain why three applications are needed.

Composite functions