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For each of the following functions : 1
MATHEMATICS · Multi-part · AoPS · Mathematics · Functions · Functions
Problem
For each of the following functions
f
f
f
: 1. Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
. 2. Sketch
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
,
y
=
f
−
1
(
x
)
y=f^{-1}(x)
y
=
f
−
1
(
x
)
, and
y
=
x
y=x
y
=
x
on the same set of axes. 3. Show that
(
f
−
1
∘
f
)
(
x
)
=
(
f
∘
f
−
1
)
(
x
)
=
x
(f^{-1}\circ f)(x)=(f\circ f^{-1})(x)=x
(
f
−
1
∘
f
)
(
x
)
=
(
f
∘
f
−
1
)
(
x
)
=
x
, the identity function. (a)
f
:
x
↦
2
x
+
5
f:x\mapsto2x+5
f
:
x
↦
2
x
+
5
(b)
f
:
x
↦
3
−
2
x
4
f:x\mapsto\frac{3-2x}{4}
f
:
x
↦
4
3
−
2
x
(c)
f
:
x
↦
x
+
3
f:x\mapsto x+3
f
:
x
↦
x
+
3
Related theory
Composition of Functions
Functions
Algebra
Composite functions
Relations and Functions
Invertible functions and inverse of a function
Exponential and Logarithmic Functions
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