Calculus- Early Transcendentals, 2021a

2.2. Transformations and Compositions 49
(a) y = x 2 + 1
(b) y = f (x)= √ 2x − 3
(c) y = f (x)=1/(x + 1)
(d) y = f (x)=1/(x 2 − 1)
(e) y = f (x)= √ −1/x
(f) y = f (x)= 3√ x
√
(g) y = f (x)= r 2 − (x − h) 2 , where r and h are
positive constants.
(h) y = f (x)= 4√ x
(i) y =
√
1 − x 2
(j) y = f (x)= √ 1 − (1/x)
√
(k) y = f (x)=1/ 1 − (3x) 2
(l) y = f (x)= √ x + 1/(x − 1)
(m) y = f (x)=1/( √ x − 1)
Exercise 2.1.2 A farmer wants to build a fence along a river. He has 500 feet of fencing and wants to
enclose a rectangular pen on three sides (with the river providing the fourth side). If x is the length of the
side perpendicular to the river, determine the area of the pen as a function of x. What is the domain of this
function?
Exercise 2.1.3 A can in the shape of a cylinder is to be made with a total of 100 square centimeters
of material in the side, top, and bottom; the manufacturer wants the can to hold the maximum possible
volume. Write the volume as a function of the radius r of the can; find the domain of the function.
Exercise 2.1.4 A can in the shape of a cylinder is to be made to hold a volume of one liter (1000 cubic
centimeters). The manufacturer wants to use the least possible material for the can. Write the surface
area of the can (total of the top, bottom, and side) as a function of the radius r of the can; find the domain
of the function.
2.2 Transformations and Compositions
2.2.1 Transformations
Transformations are operations we can apply to a function in order to obtain a new function. The most
common transformations include translations (shifts), stretches and reflections. We summarize these below.
Function Conditions How to graph F(x) given the graph of f(x)
F(x)= f (x)+c c> 0 Shift f (x) upwards by c units
F(x)= f (x) − c c> 0 Shift f (x) downwards by c units
F(x)= f (x + c) c > 0 Shift f (x) to the left by c units
F(x)= f (x − c) c > 0 Shift f (x) to the right by c units
F(x)=− f (x)
Reflect f (x) about the x-axis
F(x)= f (−x)
F(x)=| f (x)|
Reflect f (x) about the y-axis
Take the part of the graph of f (x) that lies
below the x-axis and reflect it about the x-axis