# College Algebra Questions With Solutions and Explanations - sample 1

Solutions and full explanations to the set of college algebra multiple choice questions are presented.
1. 9log9(4) =

Solution

Exponential and log functions are inverse of each other. Hence

aloga(x) = x , for all x real and positive.

and therefore

9log9(4) = 4
2. 3log3(-5) =

Solution

Since -5 is not in the domain of function log3(x),

3log3(-5) is undefined
3. If f(x) = -2x2 + 8x - 4, which of the follwoing is true?

 A. The maximum value of f(x) is -4. B. The graph of f opens upward. C. The graph of f has no x-intercept D. f is not a one to one function.

Solution

f(x) is a quadratic function and its graph is a parabola that may be intercepted by horizontal lines at two points and therefore is not a one to one function. The answer is D
4. If f(x) = 5 - 2x, then f -1(-3) =

Solution

Find f -1(x) and then Find f -1(- 3)

y = 5 - 2x , given

x = 5 - 2y , interchange x and y

2y = 5 - x , y = log2(5 - x) , solve for y

f -1(x) = log2(5 - x) , inverse function

f -1(- 3) = log2(5 + 3)

= log2(23) = 3
5. If logx(3) = 1/4, then x =

Solution

Rewrite the given equation in exponential form

logx(3) = 1/4 if and only if x(1/4) = 3

We now solve, for x, the exponential equation obtained above by raising both sides to the power 4.

(x(1/4)) 4 = 3 4

x = 3 4 = 81
6. If f(x) = -x2 + 1, then f(x + 1) =

Solution

Substitute x by x + 1 in the formula of f(x) to obtain f(x + 1).

f(x + 1) = - (x + 1) 2 + 1

Expand and simplify.

f(x + 1) = - x 2 - 2x - 1 + 1 = - x 2 - 2x
7. If f(x) = x - 4, then (f o f)(3) =

Solution

(f o f)(3) = f(f(3)) = f(3 - 4) = f(-1) = - 5
8. If ln(3x - 2) = 1, then x =

Solution

Rewrite given equation in exponential form.

ln(3x - 2) = 1 if and only if e 1 = 3x - 2

Solve e 1 = 3x - 2 for x.

x = (e + 2) / 3
9. The number of solutions of (x2 + 1)2 + 2(x2 + 1) - 3 = 0 is equal to

Solution

Let u = x2 + 1 and rewrite the given equation in terms of u as follows

u 2 + 2u - 3 = 0

Factor and solve the above equation

(u + 3)(u - 1) = 0

two solutions: u = x2 + 1 = - 3 and u = x2 + 1 = 1

Equation x2 + 1 = - 3 has no solutions. Solve the equation x2 + 1 = 1 for to get

x = 0.

The given equation has one solution.
10. If the graph of y = (x - 2)2 - 3 is translated 5 units up and 2 units to the right, then the equation of the graph obtained is given by

Solution

If the graph of y = f(x) is translated 5 units up, the equation of the new graph is given by

y = f(x) + 5

If the graph of y = f(x) + 5 is translated 2 units to the right, the equation of the new graph is given by

y = f(x - 2) + 5 = ((x - 2) - 2)2 - 3 + 5

= (x - 4)2 + 2
11. If f(x) = -ex - 2, then the range of f is given by the interval

 A. (-∞ , -2) B. (-∞ , +∞) C. (-2; , +∞) D. (-∞ , +2)

12. If f(x) = √(x - 1) / (x2 - 9), then the domain of f is given by the interval

 A. (1 , +∞) B. (-3 , +3) C. [1 , 3)U(3 , +∞) D. (-3 , 3)U(3 , +∞)

13. The number of points of intersections of the graphs of y = 2x and y = -x2 + 2 is equal to

 A. 0 B. 1 C. 2 D. 3

14. If f(x) = ln(x + 1) - 2, then f-1(x) =

 A. ex + 1 - 2 B. ex - 2 C. ex + 2 - 2 D. ex + 2 - 1

15. For all x real, √(x2 -4x + 4) =

 A. x - 2 B. x + 2x + 2 C. |x - 2| D. x + 2

16. The value of x that makes x2 + 6x + 13 maximum is equal to

 A. 6 B. -3 C. 13 D. 3

17. eln(3) - ln(2) + ln(1/x) =

 A. 3 / (2x) B. 3x/2 C. 1 + 1/x D. 3/2 - 1/x

18. If f(x) = (x - 1) / (x + 2), then the range of f is given by the interval

 A. (-∞ , -2)U(-2 , +∞) B. (-∞ , 1)U(1 , +∞) C. (-2; , +∞) D. (-∞ , 1)

19. ln((x - 1)2) = 2 ln(x - 1) for all x in the interval

 A. (-∞ , +∞) B. [0 , +∞) C. (-∞ , 1)U(1 , +∞) D. (1 , +∞)

20. Let f(x) = x2 + 2x + 4. Which of the following statements is NOT true?

 A. f(x) has a maximum value B. The graph of f is not a line C. The graph of f has no x-intercepts. D. The graph of f has a y-intercept.