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For [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) = x + 2 \)[/tex], find the following functions:

a. [tex]\((f \circ g)(x)\)[/tex]
b. [tex]\((g \circ f)(x)\)[/tex]
c. [tex]\((f \circ g)(2)\)[/tex]
d. [tex]\((g \circ f)(2)\)[/tex]

Sagot :

GinnyAnswer
Sure, let's solve each part step-by-step:

### a. [tex]\((f \circ g)(x)\)[/tex]

The composite function [tex]\((f \circ g)(x)\)[/tex] is defined as [tex]\(f(g(x))\)[/tex]. This means we need to substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex].

Given functions:
- [tex]\(f(x) = \sqrt{x}\)[/tex]
- [tex]\(g(x) = x + 2\)[/tex]

Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ (f \circ g)(x) = f(g(x)) = f(x + 2) \][/tex]

Now, substitute [tex]\(g(x) = x + 2\)[/tex] into [tex]\(f(x) = \sqrt{x}\)[/tex]:
[tex]\[ f(x + 2) = \sqrt{x + 2} \][/tex]

So, [tex]\((f \circ g)(x) = \sqrt{x + 2}\)[/tex].

### b. [tex]\((g \circ f)(x)\)[/tex]

The composite function [tex]\((g \circ f)(x)\)[/tex] is defined as [tex]\(g(f(x))\)[/tex]. This means we need to substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex].

Given functions:
- [tex]\(f(x) = \sqrt{x}\)[/tex]
- [tex]\(g(x) = x + 2\)[/tex]

Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[ (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) \][/tex]

Now, substitute [tex]\(f(x) = \sqrt{x}\)[/tex] into [tex]\(g(x) = x + 2\)[/tex]:
[tex]\[ g(\sqrt{x}) = \sqrt{x} + 2 \][/tex]

So, [tex]\((g \circ f)(x) = \sqrt{x} + 2\)[/tex].

### c. [tex]\((f \circ g)(2)\)[/tex]

We have already determined [tex]\((f \circ g)(x) = \sqrt{x + 2}\)[/tex]. Now we need to evaluate it at [tex]\(x = 2\)[/tex]:

[tex]\[ (f \circ g)(2) = \sqrt{2 + 2} = \sqrt{4} = 2 \][/tex]

So, [tex]\((f \circ g)(2) = 2\)[/tex].

### d. [tex]\((g \circ f)(2)\)[/tex]

We have already determined [tex]\((g \circ f)(x) = \sqrt{x} + 2\)[/tex]. Now we need to evaluate it at [tex]\(x = 2\)[/tex]:

[tex]\[ (g \circ f)(2) = \sqrt{2} + 2 \][/tex]

The value of [tex]\(\sqrt{2}\)[/tex] is approximately 1.414213562373095, so:

[tex]\[ \sqrt{2} + 2 \approx 1.414213562373095 + 2 = 3.414213562373095 \][/tex]

So, [tex]\((g \circ f)(2) \approx 3.414213562373095\)[/tex].

Hence, the solutions are:
- a. [tex]\((f \circ g)(x) = \sqrt{x + 2}\)[/tex]
- b. [tex]\((g \circ f)(x) = \sqrt{x} + 2\)[/tex]
- c. [tex]\((f \circ g)(2) = 2\)[/tex]
- d. [tex]\((g \circ f)(2) \approx 3.414213562373095\)[/tex]
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