24
Chapter 1 | Functions and Graphs
x = ± 1
.
y −1
Since 1/ y −1 is a real number if and only if y >1, the range of f is the set ⎧ ⎩ ⎨ y | y >1
⎫ ⎭ ⎬ .
d. ( f ∘ g )(4) = f ( g (4)) = f ⎛ ⎝ 1 4 ⎞ ⎠ = ⎛ ⎝ 1 4
⎞ ⎠
2
+1= 17 16
⎛ ⎝ − 1 2
⎞ ⎠ = f
⎛ ⎝ g
⎛ ⎝ − 1 2
⎞ ⎠ ⎞ ⎠ = f (−2) = (−2) 2 +1=5
( f ∘ g )
In Example 1.7 , we can see that ⎛
⎝ f ∘ g ⎞ ⎠ ( x ) ≠ ⎛ ⎝ g ∘ f ⎞ ⎠ ( x ). This tells us, in general terms, that the order in which we compose
functions matters.
Let f ( x ) =2−5 x . Let g ( x ) = x . Find ⎛ ⎝ f ∘ g ⎞ ⎠ ( x ).
1.5
Example 1.8 Composition of Functions Defined by Tables
Consider the functions f and g described by Table 1.4 and Table 1.5 .
−3 −2 −1
x
0 1
2 3
4
f ( x )
−2
−2
0
4
2
4
0
4
Table 1.4
−4 −2
x
0 2 4
g ( x )
1
0
3 0 5
Table 1.5
a. Evaluate ( g ∘ f )(3), ⎛ ⎞ ⎠ (0). b. State the domain and range of ⎛ ⎝ g ∘ f c. Evaluate ( f ∘ f )(3), ⎛ ⎞ ⎠ (1). d. State the domain and range of ⎛ ⎝ f ∘ f
⎞ ⎠ ( x ).
⎝ g ∘ f
⎞ ⎠ ( x ).
⎝ f ∘ f
Solution
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online