STRETCH A GRAPH VERTICAL OR HORIZONTAL EXAMPLES

The procedure for stretching the graph of a function vertically or horizontally is illustrated by the following examples :

Question 1 :

Define a function g by g(x)  =  2f(x),

where f is the function defined by f(x) = x², with the domain of f the interval [−1, 1].

(a) Find the domain of g.

(b) Find the range of g.

(c) Sketch the graph of g.

Solution :

(a)  Here the function g(x) is defined by the function f(x), so the domain of both functions will be same. Hence the domain of g in the interval [-1, 1].

(b)  By multiplying each range of f(x) by 2, we get the range of the function g(x). 

Range of f(x) by applying domain, we get

f(x) = x²

If x = -1, then y  = 1

If x = 0, then y  = 0

If x = 1, then y  = 1

Range of f(x) is [0, 1]. By multiplying 2, we get the range of g(x). That is [0, 2]. 

(c)  Since 2 is the integer multiplied with the function, we have to stretch the graph vertically.

Question 2 :

Define a function h by 

h(x) = f(2x),

where f is the function defined by f(x) = x², with the domain of f the interval [−1, 1].

(a) Find the domain of h.

(b) Find the range of h.

Solution :

(a)  The formula defining h shows that h(x) is defined precisely when f(2x) is defined, which means that 2x must be in the interval [−1, 1].

Here the constant c is 1/2, which means that x must be in the interval [−1/2 , 1/2 ]. Thus the domain of h is the interval [−1/2 , 1/2].

(b) Because h(x) equals f(2x), we see that the values taken on by h are the same as the values taken on by f . Thus the range of h equals the range of f , which is the interval [0, 1].

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