**Unformatted text preview: **ction by switching
the inputs and outputs.
Example 5.2.2. Find the inverse of the following one-to-one functions. Check your answers analytically using function composition and graphically.
1. f (x) = 1 − 2x
5 2. g (x) = 2x
1−x Solution.
1. As we mentioned earlier, it is possible to think our way through the inverse of f by recording
the steps we apply to x and the order in which we apply them and then reversing those steps
in the reverse order. We encourage the reader to do this. We, on the other hand, will practice
the algorithm. We write y = f (x) and proceed to switch x and y 300 Further Topics in Functions y = f (x)
1 − 2x
y=
5
1 − 2y
x=
5
5x = 1 − 2y
5x − 1 = −2y
5x − 1
=y
−2
y = −5x +
2 switch x and y 1
2 1
We have f −1 (x) = − 5 x+ 2 . To check this answer analytically, we ﬁrst check that f −1 ◦ f (x) =
2
x for all x in the domain of f , which is all real numbers. f −1 ◦ f (x) = f −1 (f (x))
= − 5 f (x) +
2 1
2 1 − 2x
+
5
= − 1 (1 − 2x) + 1
2
2
= −5
2 1...

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