For $$ f(x)=\sqrt{x} $$ and $$ g(x)=x+6 $$, find the following functions. a. $$ (f \circ g)(x) ; b .(g \circ f)(x) ; $$ c. $$ (f \circ g)(3) $$; d. $$ (g \circ f)(3) $$ a. $$ (f \circ g)(x)=\sqrt{x+6} $$ (Simplify your answer.) b. $$ (g \circ f)(x)=\sqrt{x}+6 $$ (Simplify your answer.) c. $$ (f \circ g)(3)=\square $$ (Simplify your answer.)

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UpStudy AI · Accepted Answer

Evaluate the expression by following steps:
- step0: Evaluate:
  $$\sqrt{x}+6$$
- step1: Substitute:
  $$\sqrt{3}+6$$
Substitute $$ x=3 $$ into the expression $$ \sqrt{x+6} $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$\sqrt{x+6}$$
- step1: Substitute:
  $$\sqrt{3+6}$$
- step2: Add the numbers:
  $$\sqrt{9}$$
- step3: Write in exponential form:
  $$\sqrt{3^{2}}$$
- step4: Simplify the root:
  $$3$$
Let's go through the functions step by step:

### a. $$ (f \circ g)(x) $$
The composition of functions $$ f $$ and $$ g $$ is defined as:
$$
(f \circ g)(x) = f(g(x))
$$
Substituting $$ g(x) = x + 6 $$ into $$ f(x) = \sqrt{x} $$:
$$
(f \circ g)(x) = f(x + 6) = \sqrt{x + 6}
$$
So, the simplified answer is:
$$
(f \circ g)(x) = \sqrt{x + 6}
$$

### b. $$ (g \circ f)(x) $$
The composition of functions $$ g $$ and $$ f $$ is defined as:
$$
(g \circ f)(x) = g(f(x))
$$
Substituting $$ f(x) = \sqrt{x} $$ into $$ g(x) = x + 6 $$:
$$
(g \circ f)(x) = g(\sqrt{x}) = \sqrt{x} + 6
$$
So, the simplified answer is:
$$
(g \circ f)(x) = \sqrt{x} + 6
$$

### c. $$ (f \circ g)(3) $$
Now, we need to evaluate $$ (f \circ g)(3) $$:
$$
(f \circ g)(3) = \sqrt{3 + 6} = \sqrt{9} = 3
$$
So, the answer is:
$$
(f \circ g)(3) = 3
$$

### d. $$ (g \circ f)(3) $$
Now, we need to evaluate $$ (g \circ f)(3) $$:
$$
(g \circ f)(3) = \sqrt{3} + 6
$$
So, the answer is:
$$
(g \circ f)(3) = \sqrt{3} + 6
$$

### Summary of Results:
- a. $$ (f \circ g)(x) = \sqrt{x + 6} $$
- b. $$ (g \circ f)(x) = \sqrt{x} + 6 $$
- c. $$ (f \circ g)(3) = 3 $$
- d. $$ (g \circ f)(3) = \sqrt{3} + 6 $$
a. $$ (f \circ g)(x) = \sqrt{x + 6} $$ b. $$ (g \circ f)(x) = \sqrt{x} + 6 $$ c. $$ (f \circ g)(3) = 3 $$ d. $$ (g \circ f)(3) = \sqrt{3} + 6 $$

For and , find the following functions.
a. c. ; d.
a.
(Simplify your answer.)
b.
(Simplify your answer.)
c.
(Simplify your answer.)

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