To solve the problem, we need to interpret the two functions provided:
- \( f(x) = x^2 - 6 \)
- \( f(2x) = 2x - 6 \)
It seems like you want to analyze the relationship between these two functions in order to determine a specific value of \( k \).
Let's break down both functions:
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For \( f(x) = x^2 - 6 \): This is a standard quadratic function that opens upwards with its vertex at the point \( (0, -6) \). The graph will intersect the y-axis at \( -6 \) and is symmetric about the y-axis.
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For \( f(2x) = 2x - 6 \): To find the equivalent function in terms of \( x \), we substitute \( 2x \) with \( x \). This means that we are looking for \( f(x) \) such that: \[ f(x) = 2\left(\frac{x}{2}\right) - 6 = x - 6 \] That means: \[ f(2x) \text{ is a linear function } x - 6 \]
Now, by analyzing both \( f(x) \) and \( f(2x) \):
- The function \( f(x) = x^2 - 6 \) is a quadratic function.
- The function \( f(2x) = 2x - 6 \) produces a linear equation (by substituting \( u = 2x \)), and we reverted it back to regular notation above.
If you're looking for a point where these functions intersect or any value \( k \) related to solutions or intersections, we can set \( f(x) \) equal to \( f(2x) \):
\[ x^2 - 6 = 2x - 6 \]
Now, simplify the equation:
\[ x^2 - 2x = 0 \]
Factor out \( x \):
\[ x(x - 2) = 0 \]
Thus, the solutions are:
\[ x = 0 \quad \text{or} \quad x = 2 \]
If \( k \) is defined in the context of these solutions or intersections between both functions, it can take several forms such as the x-coordinate of intersections or substitute back into the function values.
Finding \( k \): If we use \( k = f(0) \) and \( k = f(2) \):
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For \( x = 0 \): \[ f(0) = 0^2 - 6 = -6 \]
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For \( x = 2 \): \[ f(2) = 2^2 - 6 = 4 - 6 = -2 \]
To summarize, \( k \) could be \( -6 \) or \( -2 \) based on the function values at the intersection points. If you have a specific definition for \( k \) in your context, please clarify!