10. Higher Order Thinking Consider the functions $$ g(x)=2 x+1 $$ and $$ h(x)=2 x+2 $$ for the domain $$ 0

Question

UpStudy AI · Accepted Answer

To compare the ranges of the functions $$ g(x) = 2x + 1 $$ and $$ h(x) = 2x + 2 $$ for the domain $$ 0 < x < 5 $$, we can analyze the behavior of each function based on their definitions.

1. **Determine the range of $$ g(x) $$**:
   - The function $$ g(x) = 2x + 1 $$ is a linear function with a slope of 2. 
   - As $$ x $$ approaches 0 from the right (i.e., $$ x 	o 0^+ $$), $$ g(x) $$ approaches $$ g(0) = 2(0) + 1 = 1 $$.
   - As $$ x $$ approaches 5, $$ g(x) $$ approaches $$ g(5) = 2(5) + 1 = 11 $$.
   - Therefore, the range of $$ g(x) $$ for $$ 0 < x < 5 $$ is $$ (1, 11) $$.

2. **Determine the range of $$ h(x) $$**:
   - The function $$ h(x) = 2x + 2 $$ is also a linear function with a slope of 2.
   - As $$ x $$ approaches 0 from the right, $$ h(x) $$ approaches $$ h(0) = 2(0) + 2 = 2 $$.
   - As $$ x $$ approaches 5, $$ h(x) $$ approaches $$ h(5) = 2(5) + 2 = 12 $$.
   - Therefore, the range of $$ h(x) $$ for $$ 0 < x < 5 $$ is $$ (2, 12) $$.

3. **Compare the ranges**:
   - The range of $$ g(x) $$ is $$ (1, 11) $$.
   - The range of $$ h(x) $$ is $$ (2, 12) $$.
   - We can see that the range of $$ g(x) $$ starts at 1 and goes up to 11, while the range of $$ h(x) $$ starts at 2 and goes up to 12.
   - Thus, the range of $$ g(x) $$ is entirely below the range of $$ h(x) $$.

In conclusion, the ranges of the functions $$ g(x) $$ and $$ h(x) $$ do not overlap, and the range of $$ g(x) $$ is less than the range of $$ h(x) $$.
The range of $$ g(x) $$ is $$ (1, 11) $$ and the range of $$ h(x) $$ is $$ (2, 12) $$. The range of $$ g(x) $$ is entirely below the range of $$ h(x) $$.

Higher Order Thinking Consider the functions
and for the domain
.
a. Without evaluating or graphing the
functions, how do the ranges compare?

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions

Higher Order Thinking Consider the functions and for the domain . a. Without evaluating or graphing the functions, how do the ranges compare?