Question 1

A rectangular garden measures $$4\ 	ext{ft} 	imes 10\ 	ext{ft}$$ and is surrounded by a walkway of uniform width $$x. If $$the total area (garden plus walkway) is $$112\ 	ext{ft}^2$$, what is the width $$x of $$the walkway?

Accepted Answer

$$x = 2$$

Question 2

Solve for $$x$$ and check your answer:

$$\displaystyle \frac{x}{x-7} - \frac{7}{x+7} = \frac{14x}{x^2$-49}$$

Accepted Answer

$$x = 8$$

Question 3

Solve for $$x$$ and check your answer:

$$\sqrt{6x-11} - 2x = -7$$

Accepted Answer

$$x = 3$$

Question 4

Solve using substitution:

$$(x^2-1)^2$ - 8(x^2$-1) - 20 = 0$

Accepted Answer

$$x = \pm 3,\ x = \pm 1$$

Question 5

Solve the rational inequality using the test-point method and express your answer in interval notation:

$$\displaystyle \frac{x^2$-2x-35}{x^2$-16} \geq 0$$

Accepted Answer

$$(-\infty, -5] \cup [7, \infty)$$

Question 6

Solve the inequality and express your answer in interval notation:

$$\displaystyle \frac{1}{4}|2x+1| + 1 \geq 13$$

Accepted Answer

$$x \leq -25 or$$ $$x \geq 25$$

Question 7

Given the points $$P(-5, 0)$$ and $$Q(1, -3)$$, what is the midpoint and the distance between $$P$$ and $$Q$$?

Accepted Answer

Midpoint: $$(-2, -1.5)$$, Distance: $$\sqrt{45}$$

Question 8

Given the circle $$x^2$$ + $$y^2 + 6x + 10y = 4$, what is its standard form, center, and radius?

Accepted Answer

Standard form: $$(x+3)^2$$ + $$(y+5)^2 = 30$, Center: (-3$$, -5)$$, Radius: $$\sqrt{30}$$

Question 9

Find the domain of the function $$g(x) = \sqrt{x+11} / (3x-1)$$, expressed in interval notation.

Accepted Answer

$$[-11, \infty)$$ except $$x = \frac{1}{3}$$

Question 10

Find the average rate of change of $$f(x) = \frac{5}{4x-1}$$ from $$x = -1 to$$ $$x = 1.$$

Accepted Answer

$$-\frac{5}{3}$$

Question 11

Simplify the difference quotient $$\displaystyle \frac{f(x+h)-f(x)}{h}$$ for $$f(x) = -4x^2$ + 7x - 1.$$

Accepted Answer

$$-8x + 7 - 4h$$

Question 12

Given $$f(x) = \frac{3x+4}{x-2}$$ and $$g(x) = \frac{1}{x-2}$$, evaluate $$(fg)(6).$$

Accepted Answer

$$\frac{11}{2}$$

Question 13

For $$f(x) = \frac{3x+4}{x-2}$$ and $$g(x) = \frac{1}{x-2}$$, find $$(f \circ g)(x)$$ and state its domain.

Accepted Answer

$$(f \circ g)(x) = \frac{3/(x-2) + 4}{1/(x-2) - 2}$$, Domain: $$(-\infty, 2) \cup (2, \infty) \setminus \{2.5\}$$

Question 14

Write $$g(x) = -2x^2$ - 16x + 17 in $$standard form $$a(x-h)^2 + k$, and identify the vertex.

Accepted Answer

$$-2(x+4)^2 + 49$, Vertex: (-4$$, 49)$$

Question 15

Find the inverse function $$f^{-1}(x$$)$$ for f(x$$) = \frac{6x-5}{4x+11}$$.

Accepted Answer

$$f^{-1}(x$$) = \frac{11x+5}{6-4x}$$

Question 16

What is the domain and range of $$f(x) = \frac{6x-5}{4x+11}$$?

Accepted Answer

Domain: $$(-\infty, -\frac{11}{4}) \cup (-\frac{11}{4}, \infty)$$, Range: $$(-\infty, \frac{6}{4}) \cup (\frac{6}{4}, \infty)$$

Question 17

Consider the polynomial $$P(x) = x^2$(x^2+1)(x^2$-36)(x+2)^4. $$What is the degree of $$P(x)$$ and the multiplicity of the zero at $$x=0$?

Accepted Answer

Degree: $$8$$, Multiplicity at $$x=0$$: $$2$$

Question 18

Which of the following best describes the end behavior of $$P(x) = x^2$(x^2+1)(x^2$-36)(x+2)^4$?

Accepted Answer

As $$x 	o \infty$$, $$P(x) 	o \infty$$; as $$x 	o -\infty$$, $$P(x) 	o \infty$$

Step-by-Step Precalculus Midterm Study Guidance

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