MATH 112 Practice Test #2

Determine which two functions are inverses of each other. Show mathematical justification for your answer.
Inverse functions:__________________________

Find the inverse of the one-to-one function.
______________________________________

Graph f as a solid line and f^-1 as a dashed line in the same rectangular coordinate space. Use interval notation to give the domain
and range of f and f^-1. Include a table of at least five values for f and f^-1 below.

Write the standard form of the equation of the circle with the given center and radius.
4) (3, -9); 1 __________________________________

Find the center and the radius of the circle.
5) (x - 9)² + (y + 8)² = 25 Center: _______________ Radius: ________________

Complete the square and write the equation in standard form. Then give the center and radius of the circle.
6) x² + y² + 2x + 12y = -28
Center:____________
Radius:____________

Divide and express the result in standard form.

___________________________

For the given quadratic function , answer each of the following questions.
8) f(x) = -x² + 4x - 6

a.) Without graphing, find the vertex. vertex:_____________
b.) Does the function have a minimum or maximum value? _________________ What is that value? ___________
c.) What is the function's domain? domain:_____________
d.) What is the function's range? range:______________

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
9) f(x) = (x + 9)² - 8 vertex: ___________________

Use the the information you find in parts a-d below to sketch the graph of the quadratic function.
10) f(x) = x² - 2x - 3
a.) What is the vertex of f(x)? ___________
b.) Write the equation of the axis of symmetry. ______________
c.) Find the x-intercept(s), if any. _____________________
d.) Find the y-intercept. _____________

Find the zeros of the polynomial function. State the multiplicity of each zero and whether the graph crosses the x -axis, or touches
the x-axis and turns around, at each intercept .
11) f(x) = -x²(x + 3)(x² - 1)

Use the Leading Coefficient Test to determine the end behavior of the polynomial function .
12) f(x) = 5x³ - 3x² - 2x + 3 __________________________________

Sketch the polynomial function by using the information found in parts a -c.
13) f(x) = x³ + 8x² - x - 8

a.) Determine the graph's end behavior.
b.) Find the x-intercepts , the multiplicity , and the behavior of the graph at each.

c.) Find the y-intercept.

Use the Intermediate Value Theorem to determine whether the polynomial function has a real zero between the given integers.
Show mathematical justification for your answer and explain your findings.
14) f(x) = 7x³ - 10x - 2; between -2 and -1

Divide using long division. Determine q(x) and r(x).

q(x):__________________
r(x):___________________

Divide using synthetic division . Determine q(x) and r(x).
16) (x² + 8x + 9) ÷ (x + 5)
q(x):__________________
r(x):___________________

Use the Remainder Theorem to find the indicated function value.
17) f(x) = 3x⁴ + 10x³ + 3x² - 3x + 65; f(-2)_______________________

Use the Rational Zero Theorem to list all possible rational zeros for the given function.
18) f(x) = 7x⁴ - x² + 2 ____________________________________

Find a rational zero of the polynomial function and use it to find all the zeros of the function.
19) f(x) = x³ - 5x² + 7x + 13 zeros:__________________________

Find an nth degree polynomial function with real coefficients satisfying the given conditions.
20) n = 3; - 6 and i are zeros; f(-3) = 60
f(x) = _________________________________

Find the domain of the rational function. Write answer in interval notation.

domain:_______________________

Use the graph of the rational function shown to complete the statement.
22)

Find the vertical asymptotes, if any, of the graph of the rational function.

Vertical asymptote(s):___________________

Find the horizontal asymptote, if any, of the graph of the rational function.

Horizontal asymptote:__________________

Find the slant asymptote, if any, of the graph of the rational function.

Slant asymptote:___________________

Graph the rational function by using the information in parts a - e. Show work to the side and place answers underneath prompts.

a.) Find the vertical asymptote(s), if any.
b.) Find the horizontal asymptote, if any.
c.) Find the x-intercept(s), if any.
d.) Find the y-intercept.
e.) Make a list of additional points to be used in your graph.
f.) Graph the function using the information above.

Answer Key
Testname: MTH 112 BTZ PRACTICE TEST #2 (SECT. 1.8, 1.9, 2.1‐2.6) FA07

1) None
2) f^-1(x) = x³ - 2
3)

f domain = (0, ∞); range = (-4, ∞)
f-1 domain = (0, ∞); range = (-4, ∞)

4) (x - 3)² + (y + 9)² = 1
5) (9, -8), r = 5
6) (x + 1)² + (x + 6)² = 9
(-1, -6), r = 3

8) (2, -2)
9) (-9, -8)
10)

11) 0, touches the x-axis and turns
around;
-3, crosses the x-axis;
-1, crosses the x-axis;
1, crosses the x-axis

12) falls to the left and rises to the right

13)


14) f(-2) = -38 and f(-1) = 1; yes


17) 51
18) ± 1/7, ± 2/7, ± 1, ± 2
19) {-1, 3 + 2i, 3 - 2i}
20) f(x) = 2x³ + 12x² + 2x + 12
21) {x|x ≠ -8, x ≠ 8}
22) +∞
23) x = 2
24) y = 3
25) y = x - 13
26)