f(x) and g(x) are the two functions which are inverse to each other where their compositions are defined if and only if the following equations are true.
f o g = f[g(x)] = x
g o f = g[f(x)] = x
Let us see some examples to understand the above concept.
Example 1 :
f(x) = x + 3
g(x) = x - 3
Are f(x) and g(x) are inverse to each other ?
Solution :
To check whether f(x) and g(x) are inverse to each other, find f o g and g o f.
f o g :
f o g = f[g(x)]
= f[x + 3]
= x + 3 - 3
= x ----(1)
g o f :
g o f = g[f(x)]
= g[x - 3]
= x - 3 + 3
= x ----(2)
From (1) and (2),
f o g = g o f = x
So, f(x) and g(x) are inverse to each other.
Example 2 :
f(x) = x – 3
g(x) = 4x + 8
Are f(x) and g(x) are inverse to each other ?
Solution :
f o g = f[g(x)]
= f[4x + 8]
= 4x + 8 - 3
= 4x + 5 ≠ x
Because f o g ≠ x, we don't have to find g o f. And f(x) and g(x) are not inverse to each other.
Example 3 :
f(x) = log_{10}(x)
g(x) = 10^{x}
Are f(x) and g(x) are inverse to each other ?
Solution :
f o g :
f o g = f[g(x)]
= f[10^{x}]
= log_{10}(10^{x})
= xlog_{10}10
= x(1)
= x ----(1)
g o f :
g o f = g[f(x)]
= g[log_{10}(x)]
= 10^{log10(x)}
= x ----(2)
From (1) and (2),
f o g = g o f = x
So, f(x) and g(x) are inverse to each other.
Example 4 :
f(x) = x – 3
g(x) = -5x
Are f(x) and g(x) are inverse to each other ?
Solution :
f o g = f[g(x)]
= f [-5x]
= -5x - 3 ≠ x
Because f o g ≠ x, we don't have to find g o f. And f(x) and g(x) are not inverse to each other.
Example 5 :
f(x) = x
g(x) = 3x + 8
Are f(x) and g(x) are inverse to each other ?
Solution :
f o g = f[g(x)]
= f[3x + 8]
= 3x + 8 ≠ x
Because f o g ≠ x, we don't have to find g o f. And f(x) and g(x) are not inverse to each other.
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