Mathematics

Mathematics

Problem

For the functions f:R→Rf:\mathbb R\to\mathbb R given by f(x)=2x+3f(x)=2x+3 and g:R→Rg:\mathbb R\to\mathbb R given by g(x)=x−32g(x)=\dfrac{x-3}{2}, verify that ff and gg are inverses of each other. a. (i) Find (f∘g)(x)(f\circ g)(x). b. (ii) Find (g∘f)(x)(g\circ f)(x). c. (iii) Hence state the relation between ff and gg.
Intuitive explanation

To verify that two functions are inverses, you need to check that composing them in both orders gives you back the input unchanged—that is, f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x for all xx. Think of it this way: if ff "does something" to a number, then gg should perfectly "undo" it, and vice versa. Here, ff multiplies by 2 and adds 3, so gg should reverse those steps in reverse order.

Intuitive explanation

To verify that two functions are inverses, you need to check that composing them in both orders gives you back the input unchanged—that is, f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x for all xx. Think of it this way: if ff "does something" to a number, then gg should perfectly "undo" it, and vice versa. Here, ff multiplies by 2 and adds 3, so gg should reverse those steps in reverse order.

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