Given the following: $$ f(x)=\sqrt{x+3}, g(x)=3 x $$, and $$ h(x)=x^{2}-3 $$ determine the solution to the following composition. $$ f(h(x)) $$ $$ 3 \sqrt{x+3} $$, as long as $$ x \geq 0 $$ $$ \sqrt{3 x+3} $$, as long as $$ x \geq 0 $$ $$ x $$, as long as $$ x \geq 0 $$ $$ 3 x^{2}-3 $$, as long as $$ x \geq 0 $$

Question

UpStudy AI · Accepted Answer

To determine the composition $$ f(h(x)) $$, let's break it down step by step.

**Given Functions:**
- $$ f(x) = \sqrt{x + 3} $$
- $$ h(x) = x^2 - 3 $$

**Composition $$ f(h(x)) $$:**
1. **Substitute $$ h(x) $$ into $$ f(x) $$:**
   $$
   f(h(x)) = \sqrt{(x^2 - 3) + 3} = \sqrt{x^2} = |x|
   $$
2. **Simplify the Absolute Value:**
   $$
   |x| = 
   \begin{cases} 
      x & \text{if } x \geq 0 \
      -x & \text{if } x < 0 
   \end{cases}
   $$
   
   Since the possible answers provided specify conditions for $$ x \geq 0 $$, we focus on that interval.

3. **Final Expression for $$ x \geq 0 $$:**
   $$
   |x| = x \quad \text{when} \quad x \geq 0
   $$

**Conclusion:**
$$
f(h(x)) = x, \quad \text{as long as } x \geq 0
$$

**Therefore, the correct solution is:**

**$$ x $$, as long as $$ x \geq 0 $$**

Answer: $$ x $$, as long as $$ x \geq 0 $$
$$ f(h(x)) = x $$, as long as $$ x \geq 0 $$

Given the following: , and determine the solution to the following composition.

, as long as
, as long as
, as long as
, as long as

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Algebra Questions

Given the following: , and determine the solution to the following composition. , as long as , as long as , as long as , as long as