Q: A farmer has 500 meters of fencing and wants to enclose a rectangular plot that borders a river (no fencing along the river). If $$x is $$the width perpendicular to the river, what value of $$x$$ maximizes the area, and what is the largest area that can be enclosed?

$$x = 125$$ meters, \text{Area} = 31,250$ square meters

Q: Find the inverse function of $$f(x) = 2x + 7. $$What is the domain and range of $$f$$?

$$f^{-1}(x$$) = \frac{x - 7}{2}$$, \text{Domain: } (-\infty, \infty), \text{Range: } (-\infty, \infty)$

Question 1

Given the function $$y = \sqrt{x}$$, apply the following transformations in order: (1) reflect about the x-axis, (2) shift up 5 units, (3) shift right 9 units. What is the final equation of the transformed function?

Accepted Answer

$$y = -\sqrt{x - 9} + 5$$

Question 2

A farmer has 500 meters of fencing and wants to enclose a rectangular plot that borders a river (no fencing along the river). If $$x is $$the width perpendicular to the river, what value of $$x$$ maximizes the area, and what is the largest area that can be enclosed?

Accepted Answer

$$x = 125$$ meters, 	ext{Area} = 31,250$ square meters

Question 3

Given the graph of a polynomial function with the indicated point $$(-1, -32)$$, and the graph shows three x-intercepts at $$x = -4$$, $$x = 1$$, and $$x = 2$$, what is the smallest degree polynomial that fits the graph, and what is its equation?

Accepted Answer

$$f(x) = a(x + 4)(x + 1)(x - 1)(x - 2)$$

Question 4

Which transformation will result in the function $$y = \sqrt{x}$$ being shifted left 4 units and reflected about the y-axis?

Accepted Answer

$$y = \sqrt{-x - 4}$$

Question 5

A rectangular garden is to be constructed using 60 meters of fencing, with one side against a wall (no fencing needed there). What is the maximum area that can be enclosed?

Accepted Answer

$$450$$ square meters

Question 6

If $$f(x) = 3x + 5$$ and $$g(x) = 2x^2$ + 3$$, what is $$f(g(2))$$?

Accepted Answer

$$f(g(2)) = 21$$

Question 7

What is the domain of the function $$f(x) = \frac{1}{x^2$ - 9}$$?

Accepted Answer

$$x 
eq 3, x 
eq -3$$

Question 8

Find the inverse function of $$f(x) = 2x + 7. $$What is the domain and range of $$f$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 7}{2}$$, 	ext{Domain: } (-\infty, \infty), 	ext{Range: } (-\infty, \infty)$

Question 9

Write $$\log_2$(8x^3$$)$ as the sum and/or difference of logarithms, expressing powers as factors.

Accepted Answer

$$3\log_2$(x) + \log_2$(8)$$

Question 10

Solve for $$x$$: $$2^{x+3} = 16$

Accepted Answer

$$x = 1$$

Question 11

Given the polynomial $$f(x) = 5x^4$ - 76x^3$ + 255x^2$ - 268x + 44$$, use the Rational Zeros Theorem to find all real zeros and factor $$f$$ over the real numbers.

Accepted Answer

$$f(x) = 5(x - 1)(x - 2)(x - 4)(x - 11)$$

Question 12

If the function $$f(x) = \frac{2x + 1}{x - 3}$$, what are the vertical and horizontal asymptotes?

Accepted Answer

Vertical: $$x = 3$$, Horizontal: $$y = 2$$

Question 13

Which of the following is the correct composite function $$g(f(x)) if$$ $$f(x) = x^2$ + 1$$ and $$g(x) = 2x - 3$$?

Accepted Answer

$$g(f(x)) = 2x^2$ - 1$$

Question 14

If $$f(x) = \sqrt{x - 2}$$, what is the domain of $$f$$?

Accepted Answer

$$x \geq 2$$

Question 15

Which of the following is NOT a potential rational zero of $$f(x) = 2x^3$ - 5x^2$ + 4x - 8$$?

Accepted Answer

$$\frac{5}{2}$$

Question 16

If a function $$f is $$one-to-one, which of the following statements is true?

Accepted Answer

Each $$y$$ value corresponds to at most one $$x$$ value.

College Algebra Study Guidance: Transformations, Optimization, and Polynomial Functions

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