Use geogra to graph f of x equals a square root of x + 9 and f of 3x equals the square root of 3x + 9 based on the graph what is the value of xround your answer to the nearest whole number

Answer 1

To graph the functions \( f(x) = \sqrt{x + 9} \) and \( f(3x) = \sqrt{3x + 9} \) using GeoGebra or any graphing software, you can plot the two functions on the same set of axes.

1. Graph \( f(x) = \sqrt{x + 9} \):

   • This function is defined for \( x \geq -9 \).
   • It starts at \( f(-9) = 0 \) and increases as \( x \) increases.

2. Graph \( f(3x) = \sqrt{3x + 9} \):

   • This function is defined for \( 3x + 9 \geq 0 \), or \( x \geq -3 \).
   • The function's starting point is \( f(-3) = 0 \) and it also increases as \( x \) increases.

Analysis

The two functions are similar but shifted. To find the value of \( x \) where the two functions are equal, set them equal to each other:

\[ \sqrt{x + 9} = \sqrt{3x + 9} \]

Squaring both sides:

\[ x + 9 = 3x + 9 \]

Simplifying and solving for \( x \):

\[ x + 9 - 9 = 3x + 9 - 9 \] \[ x = 3x \] \[ 0 = 2x \] \[ x = 0 \]

Conclusion

The value of \( x \) where \( f(x) = f(3x) \) is 0.

Rounding to the nearest whole number (if necessary), we still get:

\[ \boxed{0} \]