Questions: Find f ∘ g and g ∘ f. f(x) = sqrt(x+1), g(x) = x^2 (a) f ∘ g (b) g ∘ f

Transcript text: Find $f \circ g$ and $g \circ f$. \[ f(x)=\sqrt{x+1}, \quad g(x)=x^{2} \] (a) $f \circ g$ (b) $g \circ f$

Solution

Solution Steps

To find the compositions of the functions $ f $ and $ g $, we need to substitute one function into the other. Specifically: (a) For $ f \circ g $, substitute $ g(x) $ into $ f(x) $. (b) For $ g \circ f $, substitute $ f(x) $ into $ g(x) $.

Step 1: Find $ f \circ g $

To find $ f \circ g $, we substitute $ g(x) = x^2 $ into $ f(x) = \sqrt{x + 1} $: \[ f \circ g = f(g(x)) = f(x^2) = \sqrt{x^2 + 1} \]

Step 2: Find $ g \circ f $

Next, we find $ g \circ f $ by substituting $ f(x) = \sqrt{x + 1} $ into $ g(x) = x^2 $: \[ g \circ f = g(f(x)) = g(\sqrt{x + 1}) = (\sqrt{x + 1})^2 = x + 1 \]