Verify that f and g are inverse functions of each other, where (i) f(x) = (x- 7)/4, g(x) = 4x + 7 (ii) f(x) = x^3 + 4, g(x) = √(x - 4)

Question

Verify that f and g are inverse functions of each other, where

(i) f(x) = \(\frac{x-7}4,\) g(x) = 4x + 7

(ii) f(x) = x³ + 4, g(x) = \(\sqrt[3]{x-4}\) 

(iii) f(x) = \(\frac{x+3}{x-2}, g(x) = \frac{2x+3}{x-1}\)

Answer 1

(i) f(x) = \(\frac{x-7}4\)

Replacing x by g(x), we get

f[g(x)] = \(\frac{g(x)-7}4\) = \(\frac{4x+7-7}4\) = x

g(x) = 4x + 7

Replacing x by f(x), we get

g[f(x)] = 4f(x) + 7 = \(4(\frac{x-7}4)+7\) = x

Here, f[g(x)] = x and g[f(x)] = x.

∴ f and g are inverse functions of each other.

(ii) f(x) = x³ + 4

Replacing x by g(x), we get

f[f(x)] = [g(x)³ + 4]

 = \((\sqrt[3]{x-4})^3+4\) 

= x – 4 + 4

= x

g(x) = \(\sqrt[3]{x-4}\) 

Replacing x by f(x), we get

g[f(x)] = \(\sqrt[3]{f(x)-4}=\sqrt[3]{x^3+4-4}=\sqrt[3]{x^3}\)

 = x

Here, f[g(x)] = x and g[f(x)] = x

∴ f and g are inverse functions of each other.

(iii) f(x) = \(\frac{x+3}{x-2}\) 

Replacing x by g(x), we get

Here, f[g(x)] = x and g[f(x)] = x.

∴ f and g are inverse functions of each other.