Inverse of a Function

The functions f(x)=2x-3 and f^(-1)(x)= x+32 will be shown to undo each other. To do so, it needs to be proven that f(f^(- 1)(x))=x and that f^(- 1)(f(x))=x. To start, the first equality will be proven. First, the definition of f will be used. f(f^(- 1)(x))? =x ⇕ 2f^(- 1)(x)-3? =x Now, in the above equation, x+32 will be substituted for f^(- 1)(x).

2f^(- 1)(x)-3? =x

Substitute

f^(- 1)(x)= x+3/2

2( x+3/2)-3? =x

▼

Simplify left-hand side

DenomMultFracToNumber

2 * a/2= a

(x+3)-3? =x

RemovePar

Remove parentheses

x+3-3? =x

SubTerm

Subtract term

x=x ✓

A similar procedure can be performed to show that f^(- 1)(f(x))=x.

	Definition of First Function	Substitute Second Function	Simplify
f(f^(- 1)(x))? =x	2f^(- 1)(x)-3? =x	2( x+3/2)-3? =x	x=x ✓
f^(- 1)(f(x))? =x	f(x)+3/2? =x	2x-3+3/2? =x	x=x ✓

Therefore, f and f^(- 1) undo each other. The graphs of these functions are each other's reflection across the line y=x. This means that the points on the graph of f^(- 1) are the reversed points on the graph of f.

$Graph of the linear function f(x)=2*x-3 with a point (a,b) on it and the graph of its inverse function f^{-1}(x)=x/2+3/2 with a point (b,a) on it are depicted. The line of symmetry x=y for these functions is also shown.$

Inverse of a Function

Why

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