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Math 141 - Midterm Review
Here are some practice problems to get ready for the exam.
(1) Chapter 1
(a) 1.1: Functions
(i) Write the function f (x) = 2x with domain {1, 2, 3, 4} using dots and arrows, and
using pairs of numbers.
(ii) Express the domain of (f + g)(x) and (f /g)(x) in terms of the domains of f and
g.
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(iii) If f (x) = x−1
and g(x) =
domain of (2f + 2g)(x)?
√
x, write a formula for (2f + 3g)(x). What is the
(iv) If f (x) = x2 − x, find (and simplify) and expression for the function
f (x + h) − f (x)
.
h
(b) 1.2: Graphs of functions
(i) Given a function f (x) with domain D (a collection of real numbers), the graph
of f consists of all points (x,
), where x is in
.
(ii) Which of the following curves are graphs of functions? Why?
1
2
(iii) The graph of a function f (x) is pictured below. Is it true that f (2) = 1? Is it
true that f (0) = −3? Are there any numbers x so that f (x) = 5?
(c) 1.3: Properties of Functions
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(i) Describe what feature the graph of a function f (x) must have if it has even
symmety. Write the definition of even symmety (that is, write the equation that
f (x) must satisfy if it is to be even). Repeat for f (x) odd.
(ii) Determine if the following functions are even or odd or neither:
x4
,
x2 + 1
x3 + 1,
ln x,
ln |x|,
x+
√
3
x.
(d) 1.4 Library of Functions
(i) For each of the following functions, what is the function’s domain, range, and
graph? Is the function even or odd or neither?
x, xn for n odd, xn for n even, c for c a constant,
1
x,
√
x,
√
3
x, |x|.
(ii) Sketch a graph of the following function. What is its domain?
(
−x − 3,
f (x) = 1
x,
x < −2
−1 < x < 3.
(e) 1.5 Graphical Transformations
(i) For each graphical operation, describe how to modify the formula for the function
f (x) to affect the graph of f (x) in the indicated way:
• Shift the graph up by c units.
• stretch the graph vertically by a factor of a.
• Shift the graph to the right by c units.
• Flip the graph horizontally.
• Shift the graph to the left by c units.
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• Stretch the graph horizontally by a factor of a.
• Compress the graph vertically by a factor of a.
(ii) Sketch a graph of the following function by starting with the graph of
using graphical operations:
1
X
and
−2
− 1.
3x + 2
(iii) Sketch a graph of − ln(−x).
(2) Chapter 2: Linear and Quadratic Functions
(a) 2.1: Linear Functions
(i) What is the general form of a linear function?
(ii) If f (x) = mx + b is a linear function, what is the domain and range of f ? Do
they depend on m or b?
(iii) If f (x) is linear, how many possible solutions are there to an equation f (x) = c,
where c is some constant real number? Does such an equation always have a
solution?
(iv) When is a linear function mx + b even? When is it odd?
(v) Is the following data represented by a linear function? If so, what is the function?
x f (x)
1 2.5
3
4
5 5.5
7
7
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(b) 2.3 Quadratic functions
(i) When if a function of the form ax2 + bx + c not a quadratic function?
(ii) If c is a real number and f (x) is quadratic, how many solutions could the equation
f (x) = c have? Draw pictures demonstrating each possibility.
(iii) Find roots of f (x) = x2 − x − 3 by completing the square.
(iv) Find the intersection points of the graphs of f (x) = 12 x2 − 2x − 1 and g(x) =
−x2 + 2.
(v) What is the range of f (x) = 2x2 − 4x + 5.
(c) 2.4 Properties of Quadratic Functions
(i) When is a quadratic function ax2 + bx + c even? When is it odd?
(ii) What is the minimum value taken by the function g(x) = 3x2 + 2x − 1?
(iii) What is the minimum value taken by the function h(x) = −2x2 + 50x?
(d) 2.5 Quadratic inequalities
(i) Solve −x2 + 3x − 1 ≤ 0.
(ii) Solve x2 − x + 2 < 12 x2 + 10.
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(iii) Is it possible for an inequality of the form ax2 + bx + c > 0 to have (1, 2) ∪ (3, 4)
as its set of solutions? How about [−10, 10]? How about {5}?
(e) 2.8 Equations and Inequalities involving the Absolute Value
(i) Sketch a graph of f (x) = |x + 1| − 1 and use your picture to solve the inequality
f (x) > 3.
(ii) Solve |x2 − 1| = 1.
(iii) Solve |x2 + 6x − 2| = −3.