Q: Suppose the response of a frog's heart muscle to acetylcholine is modeled by the function $$R(x) = \frac{100x}{b + x}$$, where $$x is $$the concentration of acetylcholine and $$b is a $$positive constant. If $$b = 20$$, what is the response when $$x = 60$$?

$$R(60) = \frac{100 \times 60}{20 + 60} = 75$$

Q: Given $$f(x) = x^2$ - 2x - 1$$, is the point $$(3, 12) on $$the graph of $$f$$?

No, because $$f(3) = 3^2$ - 2 \times 3 - 1 = 9 - 6 - 1 = 2$$, not 12.

Q: If $$f(x) = 2x + 1$$ and $$g(x) = x^2$, what is (f$$ \circ g)(2)$$?

$$(f \circ g)(2) = f(g(2)) = f(4) = 2 \times 4 + 1 = 9$$

Question 1

Suppose the response of a frog's heart muscle to acetylcholine is modeled by the function $$R(x) = \frac{100x}{b + x}$$, where $$x is $$the concentration of acetylcholine and $$b is a $$positive constant. If $$b = 20$$, what is the response when $$x = 60$$?

Accepted Answer

$$R(60) = \frac{100 	imes 60}{20 + 60} = 75$$

Question 2

Given the linear function $$y = Kx + V$$, if $$y = 50$$ when $$x = 0$$ and $$y = 0.2$$ when $$x = 0.02$$, what are the values of $$K$$ and $$V$$?

Accepted Answer

$$K = 0.004,\ V = 0.02$$

Question 3

Which of the following intervals is a closed interval on the real number line?

Accepted Answer

$$[2, 5]$$

Question 4

A piecewise function is defined as $$f(x) = 2x + 1$$ for $$x \leq 3$$ and $$f(x) = x^2$ for x$$ \u003e 3$$. What is f(3$$)$$?

Accepted Answer

$$f(3) = 2 	imes 3 + 1 = 7$$

Question 5

Given $$f(x) = \frac{1}{x}$$ and $$g(x) = \frac{2}{x+1}$$, what is $$(f + g)(x)$$ expressed as a single rational function?

Accepted Answer

$$(f + g)(x) = \frac{x+1 + 2x}{x(x+1)} = \frac{3x+1}{x(x+1)}$$

Question 6

If the cost function for a florist shop is $$C(x) = 7x + 80$$ and the revenue function is $$R(x) = 21x$$, what is the profit when 120 sales are made in a week?

Accepted Answer

Question 7

Which of the following statements about the vertical line test is correct?

Accepted Answer

A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Question 8

A $$500 investment earns interest compounded quarterly at an annual rate r$. Which expression gives the value after one year?

Accepted Answer

$$500 \left(1 + \frac{r}{4}\$$right)^4$$

Question 9

Given $$f(x) = x^2$ - 2x - 1$$, is the point $$(3, 12) on $$the graph of $$f$$?

Accepted Answer

No, because $$f(3) = 3^2$ - 2 	imes 3 - 1 = 9 - 6 - 1 = 2$$, not 12.

Question 10

If $$f(x) = 2x + 1$$ and $$g(x) = x^2$, what is (f$$ \circ g)(2)$$?

Accepted Answer

$$(f \circ g)(2) = f(g(2)) = f(4) = 2 	imes 4 + 1 = 9$$

Question 11

A rectangular corral with two partitions has area $$2500$$ square feet. If the outside dimensions are $$x$$ and $$y$$, what is the equation for the area?

Accepted Answer

$$xy = 2500$$

Question 12

Given the cost of fencing the boundary of a corral is $$10$$ per foot and the inner partitions cost $$8$$ per foot, with outside dimensions $$x$$ and $$y$$, what is the total cost expression?

Accepted Answer

$$10(2x + 2y) + 8(2x)$$

Question 13

If $$f(x) = x^2$ and g(x) = x + 1$, what is $$\frac{f}{g}(x)$$?

Accepted Answer

$$\frac{x^2$}{x + 1}$$

Question 14

Which of the following is the correct formula for the compound amount $$A$$ after $$t$$ years, with $$m$$ compounding periods per year and annual rate $$r$$?

Accepted Answer

$$A = P \left(1 + \frac{r}{m}\$$right)^{mt}$$

Question 15

If the surface area of a box with dimensions $$x$$, $$y$$, and $$z is $$given by $$2yz + 2xz + xy$$, which side is represented by $$xy$$?

Accepted Answer

The bottom of the box

Question 16

Given $$f(x) = x^2$ - 2x - 1$$ and $$g(x) = x$$^{1/3}$$, which function is not defined for all real numbers?

Accepted Answer

Both $$f(x)$$ and $$g(x)$$ are defined for all real numbers.

Question 17

If $$f(x) = x^2$ - 2x - 1$$ and $$g(x) = x$$^{1/3}$$, what is the domain of h(x) = f(x) + g(x$$)$$?

Accepted Answer

All real numbers

Chapter 0: Functions – Foundations for Business Calculus

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