Question 1

Suppose $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ + 3. $$What is $$(f \circ g)(x)$$ and what is its domain?

Accepted Answer

$$(f \circ g)(x) = \sqrt{x^2$ + 3}$$,\ domain:\ \mathbb{R}$

Question 2

Given $$f(x) = x^2$ and g(x) = 2x$, compute (f$$ \circ g)(x)$$ and (g$$ \circ f)(x)$$. Which of the following is true?

Accepted Answer

$$(f \circ g)(x) = 2x^2$\ and\ (g$$ \circ f)(x) = $$4x^2$$

Question 3

Let $$F(x) = (3x - 5$$)^{100}$$. Using the chain rule, what is F$$'(x)$$?

Accepted Answer

$$F'(x) = 100(3x - 5$$)^{99}$$ \cdot 3$$

Question 4

If $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$, what is the domain of (g$$ \circ f)(x)$$?

Accepted Answer

$$[0, \infty)$$

Question 5

Given $$F(x) = \frac{1}{\sqrt{2x + 3}}$$, express $$F(x) as a $$composition of four functions.

Accepted Answer

$$F(x) = (i \circ h \circ g \circ f)(x)$$\ where\ $$f(x) = 2x$$, $$g(x) = x + 3$$, $$h(x) = \sqrt{x}$$, $$i(x) = \frac{1}{x}$$

Question 6

If $$F(x) = f(g(x))$$ and both $$f$$ and $$g$$ are differentiable, what is the correct formula for $$F'(x)$$?

Accepted Answer

$$F'(x) = f'(g(x)) \cdot g'(x)$$

Question 7

Suppose $$f(x) = \frac{1}{x^2$}$$ and $$g(x) = x^2$ + x + 1. $$Find $$F'(1) if$$ $$F(x) = f(g(x)).$$

Accepted Answer

$$F'(1) = -\frac{1}{3}$$

Question 8

Let $$F(x) = 3 \ln(x^2$ + 1). $$Express $$F(x) as a $$composition of four functions.

Accepted Answer

$$F(x) = (i \circ h \circ g \circ f)(x)$$\ where\ $$f(x) = x$$, $$g(x) = x^2$ + 1$$, $$h(x) = \ln(x)$$, $$i(x) = 3x$$

Question 9

If $$F(x) = [g(h(x))]^r$$ with $$r \in \mathbb{R}$$, what is $$\frac{d}{dx} F(x)$$?

Accepted Answer

$$\frac{d}{dx} F(x) = r [g(h(x))]^{r-1} \cdot g'(h(x)) \cdot h'(x)$$

Question 10

A balloon is being inflated at a constant rate of $$1\ 	ext{ft}^3/	ext{sec}. $$Its shape is always a sphere. How fast is the radius increasing when it equals $$1$$ foot? (Recall $$V = \frac{4}{3}\pi r^3$.)

Accepted Answer

$$\frac{dr}{dt} = \frac{1}{4\pi}$$\ ft/sec

Question 11

Given $$f(x) = x^2$ + 1$$ and $$g(x) = \sqrt{x}$$, what is the domain of $$(g \circ f)(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 12

If $$F(x) = f(g(h(x)))$$, which of the following correctly expresses $$F'(x)$$ using the chain rule?

Accepted Answer

$$F'(x) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$$

Question 13

Suppose $$f(x) = x^2$, g(x) = x + 1$, and h(x$$) = \ln(x)$$. Express F(x) = h(g(f(x$$)))$$ as a composition.

Accepted Answer

$$F(x) = \ln((x^2$) + 1)$$

Question 14

Given $$f(x) = x^2$ + 1$$ and $$g(x) = \sqrt{x}$$, what is $$(g \circ f)(x)$$?

Accepted Answer

$$(g \circ f)(x) = \sqrt{x^2$ + 1}$$

Question 15

If $$F(x) = (x^2$ + x + 1$$)^{10}$$, what is F$$'(x)$$?

Accepted Answer

$$F'(x) = 10(x^2$ + x + 1)^9$ (2x + 1)$$

Question 16

Suppose $$f(x) = x^3$ and g(x) = 2x + 1$. What is (f$$ \circ g)(x)$$?

Accepted Answer

$$(f \circ g)(x) = (2x + 1)^3$

Question 17

If $$F(x) = \frac{1}{\sqrt{x^2$ + 2}}$$, find $$F'(x).$$

Accepted Answer

$$F'(x) = -\frac{x}{(x^2$ + 2$$)^{3/2}$$}$$

Question 18

Given $$f(x) = x^2$ and g(x) = x + 1$, which of the following is true about (f + g)(x$$)$$ and (f$$ \circ g)(x)$$?

Accepted Answer

$$(f + g)(x) = x^2$ + x + 1$$\ and\ $$(f \circ g)(x) = (x + 1)^2$

Composition of Functions and the Chain Rule in Business Calculus

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