MATH 121: Extra Practice for Test 1
Disclaimer: Any material covered in class and/or assigned for homework is fair game for the exam.
1. True or False:
(a) Every relation is a function.
(b) The graph of a function can be symmetric with respect to the x-axis.
(c) An element from the range of the function can be related to more then one element in the domain.
2. Determine whether the given relations are functions. Explain why or why not? If the relation is a
function state its domain and range.
(a) {(1, 1), (2, 1), (3, 1), (4, 1)}
(b) {(−1, 2), (1, 2), (3, 1), (1, 3)}
3. Determine whether the equation defines y as a function of x. Explain!
(a) x + y = 1
(c) x2 + y = 3
(b) x + y 2 = 1
(d) x2 + (y − 1)2 = 3
4. Determine if each of the following functions is even, odd or neither:
(a) f (x) =
2x
|x|
(b) g(x) = x3 − x + x7
(c) h(x) = 2x2 − 3x4 + 7
(d) F (x) = x3 − 3x + 1
(e) G(x) =
x3
x2 − 4
(f) H(x) = 3
5. Sketch a graph of a function with the following characteristics:
• Domain: [−7, ∞)
• Increasing on the open intervals (−7, 0) and (4, ∞)
• Decreasing on the open interval (0, 4)
• f (−7) = 4, f (0) = 9, and f (4) = 2.
What are the local maxima and minima?
6. Find the domain of each of the functions below and represent it using both interval and set notation.
(a) f (x) = 7x − 3
(b) h(x) =
(c) g(x) =
√
7x − 3
1
7x − 3
1
(d) p(x) = √
7x − 3
7. Consider the functions f (x) =
x
(e) r(x) = 2
x −1
1
(f) F (x) = 2
x +1
(g) G(x) =
−117x
x3 − 4x
√
√
1
and g(x) = 3 − x + x + 1.
x
(a) Find (f + g)(x).
(b) What is the domain of (f + g)(x).
(h) H(x) =
7x − 3
17
(i) P (x) = (x − 1)−2/3
(j) Q(x) = (x − 1)−3/2
(k) R =
17x + 1
√
1− x+2
g
(x).
(c) Find and simplify
f
g
(d) What is the domain of
(x).
f
8. Consider the function f (x) =
x2 − 2
.
x−4
(a) Is the point (1, 32 ) on the graph of f ?
(b) If x = 0, what is f (x)?
(c) What is f (4)? What can you say about
x = 4?
9. If f (x) =
(d) If f (x) = 12 , what is x?
(e) What is the domain of f ?
(f) List the x and y-intercepts, if any.
2x − A
, and f (4) = 0, what is the value of A? Where is f not defined?
x−3
10. For each of the functions f (x) = −3x2 + 2x and f (x) =
(a) 2f (1) + f (0)
(d) f (7x + 5)
(b) f (a2 )
(e) f (x) + h
(c) 2f (x)
(f) f (x + h)
2
evaluate the following:
x+3
(g) f (x + h) − f (x)
(h)
f (x + h) − f (x)
, h 6= 0
h
√
√
11. Sketch the graphs of the basic functions f (x) = mx + b, f (x) = x2 , f (x) = x3 , f (x) = x, f (x) = 3 x,
f (x) = |x|, f (x) = x1 . In each case, state the domain, the range, any intercepts, intervals where the
function is increasing or decreasing, local maxima or minima and determine if the function is even,
odd or neither.
12. For each of the piecewise defined functions below, state the domain, the range, the intercepts and
give a sketch:
(
(a) f (x) =
1
x
√
x
if x < 0
if x ≥ 0.


2x + 5
(b) f (x) = −2

 2
x −1
, −4 < x < −1
, −1 ≤ x < 0
, 0<x ≤3
(
−|x| + 1
(c) f (x) =
x−1
(
2x
(d) f (x) =
3
,x < 1
,x ≥ 1
, x 6= 1
,x = 1
Also, evaluate f (−2),f (0),f (8) for each function.
13. Suppose that the graph of the function y = f (x) is known. Describe how you would obtain the graph
of:
(a) y = f (x − 2) + 1
(c) y = f (4x)
(b) y = 4f (x)
(d) y = −f (−x)
14. Sketch the graphs of the following functions, starting with the graph of the basic function:
(a) f (x) =
√
−x + 2
4
+2
x−1
√
(c) g(x) = 4 x − 2 − 1
(b) f (x) =
√
(g) f (x) = − −x
1
(x − 1)2 − 3
2
√
(e) h(x) = − x + 3
(h) h(x) = −|x + 3| + 2
√
(f) h(x) = −3 3 x + 1
(i) h(x) =
(d) k(x) =
√
3−x
15. Solve the following inequalities. Write the solution set using interval notation.
(a)
1
2x
− 3 ≥ 41 x + 2
(b) −4 ≤ 3x − 1 < 5
(c) |3x + 2| < 7
(d) |x + 3| ≥ 5