To solve for the value of \( k \) using the functions you provided, let’s first clarify the functions you are interested in:
- \( f(x) = \sqrt{x + 9} \)
- \( f(3x) = \sqrt{3x + 9} \)
Next, let's analyze these functions.
- The function \( f(x) \) is defined for \( x \geq -9 \) since you cannot take the square root of a negative number without getting complex results.
- The function \( f(3x) \) suggests that you are evaluating \( f \) at \( 3x \), meaning we need to think about the input as being scaled by 3.
To graph these functions and make comparisons, it would help to evaluate them at some strategic points:
For \( f(x) = \sqrt{x + 9} \):
- At \( x = 0 \): \( f(0) = \sqrt{0 + 9} = 3 \)
- At \( x = 9 \): \( f(9) = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \)
For \( f(3x) = \sqrt{3x + 9} \):
- At \( x = 0 \): \( f(3 \cdot 0) = \sqrt{3 \cdot 0 + 9} = 3 \)
- At \( x = 3 \): \( f(3 \cdot 3) = \sqrt{3 \cdot 3 + 9} = \sqrt{18} \approx 4.24 \)
Next, since you are likely trying to find the relationship between these two functions using transformations (scaling in this case), we recognize that \( f(3x) \) compresses the input by a factor of 3.
To find \( k \), which might signify a scaling factor or a relationship based on matching \( f(x) \) and \( f(3x) \), we need to find where these two graphs might intersect or how they relate:
- The transformed function \( f(3x) \) will reach the same value, but faster due to the input being scaled.
Assuming you are looking for intersections or equivalent values in terms of \( y \), you can set: \[ \sqrt{x + 9} = \sqrt{3x + 9} \]
Squaring both sides gives: \[ x + 9 = 3x + 9 \] This leads to: \[ x = 0 \] This means at \( x = 0 \), both functions give the same value of 3.
Finally, if you are trying to solve for \( k \) related to some other value (possibly an intersection or horizontal translation), you will likely be looking for a specific correspondence between these two functions, \( k \) could represent a shift or scaling factor.
In the absence of further detail about \( k \), if based purely on equal outputs examined it is not directly evident a specific value arises without further context.
Assuming \( k \) should be a rounded output of some evaluations, a likely inference could be rounding \( \sqrt{18} \) which is \( 4.24 \rightarrow \) rounds to \( 4 \).
If there are further points of clarification regarding \( k \)’s initial value intention, I would be happy to revise the response accordingly!