Q: A medical dosage $$D(t) is $$given by $$D(t) = f(g(t))$$, where $$g(t) is $$the patient's weight in kg as a function of time $$t$$, and $$f(w) = 0.1w^2$. If g(t) = 50 + 2t$, what is D(3$$)$$?

$$f(g(3)) = 0.1 \times (56)^2$ = 313.6$$

Q: A chemical reaction rate $$R(C)$$ depends on concentration $$C as$$ $$R(C) = \ln(C + 1). If$$ $$C(t) = 2t$$, what is $$R(C(4))$$?

$$\ln(2 \times 4 + 1) = \ln(9)$$

Question 1

Given $$f(x) = x^2$ + 1$$ and $$g(x) = 2x - 3$$, find $$(f \circ g)(2). $$What is the value?

Accepted Answer

Question 2

Let $$f(x) = \sqrt{x}$$ and $$g(x) = x - 4. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \geq 4$$

Question 3

Suppose $$f(x) = \dfrac{1}{x-2}$$ and $$g(x) = x^2$. What is the domain of (f$$ \circ g)(x)$$?

Accepted Answer

$$x 
eq \pm \sqrt{2}$$

Question 4

If $$f(x) = x + 5$$ and $$g(x) = 3x$$, find $$(g \circ f)(x).$$

Accepted Answer

$$3(x + 5)$$

Question 5

Given $$f(x) = \dfrac{1}{x}$$ and $$g(x) = x - 1$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x 
eq 1$$

Question 6

Let $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$. What is the domain of (f$$ \circ g)(x)$$?

Accepted Answer

$$x \geq 0$$

Question 7

Suppose $$f(x) = \dfrac{1}{x+4}$$ and $$g(x) = \dfrac{1}{x-1}. $$What values must be excluded from the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x = 1$$ and $$x = -3$$

Question 8

If $$f(x) = x^3$ and g(x) = 2x + 1$, find (f$$ \circ g)(x)$$.

Accepted Answer

$$(2x + 1)^3$

Question 9

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

Accepted Answer

$$x \leq -\sqrt{2} or$$ $$x \geq \sqrt{2}$$

Question 10

Suppose $$f(x) = x^2$ and g(x) = x + 3$. If h(x) = (f$$ \circ g)(x)$$, which of the following is h(x$$)$$?

Accepted Answer

$$(x + 3)^2$

Question 11

Let $$f(x) = \dfrac{1}{x}$$ and $$g(x) = x^2$ - 9. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x 
eq 3$$ and $$x 
eq -3$$

Question 12

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, find (g$$ \circ f)(x)$$ and its domain.

Accepted Answer

$$\sqrt{x^2$}$$,\ x \in \mathbb{R}$

Question 13

Suppose $$f(x) = x^2$ + 1$$ and $$g(x) = \dfrac{1}{x}. $$What is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x 
eq 0$$

Question 14

Given $$f(x) = \dfrac{1}{x}$$ and $$g(x) = \dfrac{1}{x}$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$x$$

Question 15

If $$f(x) = x^2$ and g(x) = x + 1$, show that (f$$ \circ g)(x) = (g \circ f)(x)$$ for all x$ in the domain.

Accepted Answer

False, because $$(f \circ g)(x) = (x + 1)^2$ and (g$$ \circ f)(x) = $$x^2 + 1$.

Question 16

Find functions $$f$$ and $$g$$ such that $$H(x) = (3x + 2$$)^{21}$$ and H(x) = (f$$ \circ g)(x)$$. Which pair works?

Accepted Answer

$$f(x) = x$$^{21}$$,\ g(x) = 3x + 2$$

Question 17

Suppose $$H(x) = \dfrac{1}{x^2$ + 1}. $$Find $$f$$ and $$g$$ such that $$H(x) = (f \circ g)(x).$$

Accepted Answer

$$f(x) = \dfrac{1}{x},\ g(x) = x^2$ + 1$$

Question 18

A medical dosage $$D(t) is $$given by $$D(t) = f(g(t))$$, where $$g(t) is $$the patient's weight in kg as a function of time $$t$$, and $$f(w) = 0.1w^2$. If g(t) = 50 + 2t$, what is D(3$$)$$?

Accepted Answer

$$f(g(3)) = 0.1 	imes (56)^2$ = 313.6$$

Question 19

A chemical reaction rate $$R(C)$$ depends on concentration $$C as$$ $$R(C) = \ln(C + 1). If$$ $$C(t) = 2t$$, what is $$R(C(4))$$?

Accepted Answer

$$\ln(2 	imes 4 + 1) = \ln(9)$$

Composite Functions: Definitions, Domains, and Examples

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