compare the graphs of f (x) = sq rt x+ 1 - 2 and g (x) = 3 sq rt x + 1 - 2 . Identify which of the following features is true about the graphs .

Answer 1

To compare the graphs of the functions \( f(x) = \sqrt{x + 1} - 2 \) and \( g(x) = 3\sqrt{x} + 1 - 2 \), let's simplify and analyze their characteristics:

1. Domains:

   • For \( f(x) \), the function \( \sqrt{x + 1} \) requires \( x + 1 \geq 0 \), which implies \( x \geq -1 \). Therefore, the domain of \( f(x) \) is \( [-1, \infty) \).
   • For \( g(x) \), the function \( \sqrt{x} \) requires \( x \geq 0 \). Thus, the domain of \( g(x) \) is \( [0, \infty) \).

2. Intercepts:

   • For \( f(x) \):

     • y-intercept: Set \( x = 0 \) -> \( f(0) = \sqrt{1} - 2 = 1 - 2 = -1 \). So, the y-intercept is \( (0, -1) \).
     • x-intercept: Set \( f(x) = 0 \) -> \( \sqrt{x + 1} - 2 = 0 \) -> \( \sqrt{x + 1} = 2 \) -> \( x + 1 = 4 \) -> \( x = 3 \). So, the x-intercept is \( (3, 0) \).

   • For \( g(x) \):

     • y-intercept: Set \( x = 0 \) -> \( g(0) = 3\sqrt{0} + 1 - 2 = 1 - 2 = -1 \). So, the y-intercept is also \( (0, -1) \).
     • x-intercept: Set \( g(x) = 0 \) -> \( 3\sqrt{x} - 1 = 2 \) -> \( 3\sqrt{x} = 2 \) -> \( \sqrt{x} = \frac{2}{3} \) -> \( x = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \). So, the x-intercept is \( \left(\frac{4}{9}, 0\right) \).

3. Behavior:

   • \( f(x) \) starts at \( (-1, 0) \) and approaches infinity as \( x \) increases. The function is defined for \( x \geq -1 \).
   • \( g(x) \) starts at \( (0, -1) \) and also approaches infinity as \( x \) increases, defined for \( x \geq 0 \).

Comparison:

• Both functions have the same y-intercept at \( (0, -1) \).
• \( f(x) \) has an x-intercept at \( (3, 0) \) while \( g(x) \) has an x-intercept at \( \left(\frac{4}{9}, 0\right) \).
• The growth rate of \( g(x) \) is more pronounced than \( f(x) \) due to the multiplication by 3 in the graph of \( g(x) \), which means \( g(x) \) will increase faster for larger values of \( x \).

Conclusion:

• Both functions intersect at the y-axis at the same point, \( (0, -1) \).
• The key features to note are that \( f \) and \( g \) behave similarly but with different slopes and intercepts, and \( g \) grows faster due to its coefficient on the square root term.