Sure, I'd be glad to explain.
Let's first look at f(kx).
1. In the function f(kx), 'k' is a constant that multiplies the input 'x' of the function f. This operation is known as a horizontal scaling or transformation of the function.
2. The value of 'k' in f(kx) determines the degree of horizontal scaling. Specifically:
a. If 'k' is greater than 1, it contracts or compresses the function horizontally. The greater the value of 'k', the more significant the compression is.
b. If 'k' is a fraction between 0 and 1 (0 < 'k' < 1), it stretches or dilates the function horizontally. The closer the value 'k' is to 0, the more significant the stretching is.
3. This effect is called "horizontal" because it changes the way the function spreads along the x-axis.
Now, let's discuss kf(x).
4. In the function kf(x), 'k' is a constant that multiplies the output of the function f(x). This operation is known as vertical scaling or transformation of the function.
5. The value of 'k' in kf(x) has the following impact on the function:
a. If 'k' is greater than 1, it stretches or dilates the function vertically. The greater the value of 'k', the taller or more elongated the function appears.
b. If 'k' is a fraction between 0 and 1 (0 < 'k' < 1), it compresses or shrinks the function vertically. The closer 'k' gets to 0, the more the function is squashed along the vertical axis or shrinks.
6. This effect is called "vertical" because it changes the function's height along the y-axis.
In conclusion, 'k' can influence the horizontal and vertical scaling of a function, depending on whether it's part of the function's input (f(kx)) or output (kf(x)). Its exact impact depends on the value of 'k', with values between 0 and 1 creating opposite effects to values greater than 1.