Compare the graphs of f(x)=2x+1−−−−√

Question

and g(x)=2x+1−−−−√3
. Which of the following statements about the graphs is true?
Option #1: They have the same domain.

Option #2: They have the same range.

Option #3: They have the same x-intercept.

Option #4: The function f(x)
is greater than g(x)
for all values of x.

(1 point)
Option #
is the true statement about the graphs.

Answer 1

To compare the graphs of the functions \( f(x) = \sqrt{2x + 1} \) and \( g(x) = \frac{\sqrt{2x + 1}}{3} \), let's analyze each option step by step.

1. Domain:

   • For \( f(x) = \sqrt{2x + 1} \), we require \( 2x + 1 \geq 0 \) which leads to \( x \geq -\frac{1}{2} \).
   • For \( g(x) = \frac{\sqrt{2x + 1}}{3} \), the same requirement applies, \( 2x + 1 \geq 0 \) gives \( x \geq -\frac{1}{2} \).
   • Conclusion: Both functions have the same domain, \( x \geq -\frac{1}{2} \).

2. Range:

   • The range of \( f(x) \) is \( [0, \infty) \) because the square root function outputs non-negative values.
   • The range of \( g(x) \) is \( [0, \infty) \) as well, but since it is multiplied by \(\frac{1}{3}\), the output values are scaled down. Thus, the range is \( [0, \infty) \) but effectively stretched down to \( [0, \frac{1}{3} \cdot \infty) = [0, \infty) \).
   • Conclusion: They do not have the same range.

3. X-intercept:

   • To find the x-intercept for \( f(x) \), set \( f(x) = 0 \): \[ \sqrt{2x + 1} = 0 \Rightarrow 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}. \]
   • For \( g(x) \): \[ \frac{\sqrt{2x + 1}}{3} = 0 \Rightarrow \sqrt{2x + 1} = 0 \Rightarrow 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}. \]
   • Conclusion: Both functions have the same x-intercept at \( x = -\frac{1}{2} \).

4. Comparison of \( f(x) \) and \( g(x) \):

   • We note that \( g(x) = \frac{f(x)}{3} \).
   • Thus, for all \( x \geq -\frac{1}{2} \), since \( \sqrt{2x + 1} \geq 0 \), it follows that \( f(x) > g(x) \) for all \( x \).
   • Conclusion: \( f(x) \) is greater than \( g(x) \) for all values of \( x \geq -\frac{1}{2} \).

Final Conclusions:

• The functions have the same domain.
• The functions do not have the same range.
• The functions have the same x-intercept.
• The functions do meet the requirement that \( f(x) > g(x) \) for all valid \( x \).

Thus, the statement that is true about the graphs is:

Option #1: They have the same domain.

However, if you are considering only the options presented, Option #3 (They have the same x-intercept) is also correct, but based on the options given, if you need to pick one, it might be clearer to stick with their same domain. Hence, my final answer is:

Option #1 (They have the same domain).