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Math 142 © Roberto Barrera, Fall 2015
A.8 Supplement: Basic Functions and Transformations
Basic Functions
Example: Graph and find the domain and range of each of the six basic functions.
Source: Calculus for Business, Economics, Life Sciences,
and Social Sciences, 12th ed., by Barnett, Ziegler, and Byleen.
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g(x) = x2
D = (−∞, ∞)
R = [0, ∞)
f (x) = x
D = (−∞, ∞)
R = (−∞, ∞)
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m(x) = x3
D = (−∞, ∞)
R = (−∞, ∞)
√
n(x) = x
D = [0, ∞)
R = [0, ∞)
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Math 142 © Roberto Barrera, Fall 2015
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p(x) = x1/3 =
D = (−∞, ∞)
R = (−∞, ∞)
q(x) = |x|
D = (−∞, ∞)
R = [0, ∞)
√
3
x
Definitions of Some Special Functions
Power Function - A function of the form f (x) = kxr , where k and r are any real numbers. For example, f (x) = x2
is a power function.
Question: Is f (x) =
√
x a power function? Why or why not?
Solution: Yes, f (x) =
√
x = x1/2 and 1/2 is a reall number!
Polynomial Function - A function of degree n of the form f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , where an 6= 0,
an , an−1 , ...,a1 , a0 are constants, and n is a nonnegative integer. The coefficient an is called the leading coefficient,
and the domain is all real numbers. For example, f (x) = 4x3 + 2x2 + 7 is a polynomial.
Question: Is f (x) = 3x4 + 5x3 + 5x1/3 + 2 a polynomial? Why or why not?
Solution: No, all of the exponents must be nonnegative integers. 1/3 is not an integer.
f (x)
, where f (x) and g(x) are polynomial functions. The
g(x)
x2 + x + 3
domain of R(x) is all real numbers for which g(x) 6= 0. For example, R(x) =
is a rational function.
x2 − 1
Rational Function - A function of the form R(x) =
Question: What is the domain of the rational function given above?
Solution: The only restriction is that x2 − 1 6= 0 because we cannot have 0 in the denominator. So
x2 − 1 6= 0 ⇒ (x + 1)(x − 1) 6= 0 ⇒ x 6= −1, x 6= 1
and so the domain is
(−∞, −1) ∪ (−1, 1) ∪ (1, ∞).
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Math 142 © Roberto Barrera, Fall 2015
Transformations of Functions
Vertical and Horizontal Shifts and Reflections: Suppose c > 0, then
• y = f (x) + c shifts the graph of f (x) up c units
• y = f (x) − c shifts the graph of f (x) down c units
• y = f (x − c) shifts the graph of f (x) to the right c units
• y = f (x + c) shifts the graph of f (x) to the left c units
• y = − f (x) reflects the graph of f (x) about the x-axis
√
Example: Graph the function y = − x + 3 − 2 by hand, not by plotting points, but by starting with the graph of
one of the basic functions discussed above, and then applying transformations.
Solution: Basic function to start with is f (x) =
the graph 3 units to the left:
√
x. Strategy is apply what is “closest” to x. We begin by shifting
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f (x)
f (x + 3) =
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√
x+3
Then we reflect about the x-axis and then shift the graph down 2 units.
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√
− f (x + 3) = − x + 3
2
√
− f (x + 3) − 2 = − x + 3 − 2 = y
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Math 142 © Roberto Barrera, Fall 2015
Note: Always incorporate vertical shift last!
Vertical Expansion and Contraction: Suppose c > 0, then
• If c > 1, then y = c f (x) stretches (expands) the graph of f (x) vertically.
• If 0 < c < 1, then y = c f (x) compresses (contracts) the graph of f (x) vertically.
5x2
, write the function g(x) which results from shifting f (x) 3 units to the right,
x+2
then stretching vertically by a factor of 4, then reflecting about the x axis, and then shifting down 3/4 units.
Example: Starting with f (x) =
2
2
5(x−3)
(1) f (x − 3) = 5(x−3)
x−3+2 = x−1 = p(x)
2
2
20(x−3)
(2) 4p(x) = 4 · 5(x−3)
=
= q(x)
x−1
x−1
2
= m(x)
(3) −q(x) = −20(x−3)
x−1
2
(4) m(x) − 34 = −20(x−3)
− 34 = g(x)
x−1