Give two different combinations of transformations that would transform f(x)=5x+3 into g(x)=15x-12

g(x) = 15x + 9 - 9-12

= 3(5x+3) - 21
= 3 f(x) - 21

or

g(x) = 15x + 15 - 15 - 12
= 5(3x +3) - 15 - 12
= 5 f(x) - 27

To transform f(x)=5x+3 into g(x)=15x-12, we can use two different combinations of transformations:

Combination 1:
1. Horizontal Stretch: Multiply the input (x) of f(x) by a factor of 3, which will stretch the graph horizontally. The function becomes f(3x) = 15x + 3.
2. Vertical Stretch: Multiply the output (y) of f(3x) by a factor of 5, which will stretch the graph vertically. The function becomes 5[f(3x)] = 15x + 15.

Combination 2:
1. Vertical Stretch: Multiply the output (y) of f(x) by a factor of 3, which will stretch the graph vertically. The function becomes 3f(x) = 15x + 9.
2. Vertical Translation: Subtract 21 from the output (y) of 3f(x), which will shift the graph downward. The function becomes 3f - 21 = 15x - 12.

Note: These are just two examples of combinations that would transform f(x)=5x+3 into g(x)=15x-12. There may be other valid combinations as well.

To transform the function f(x)=5x+3 into g(x)=15x-12, we can use the following combinations of transformations:

Combination 1:
Start with f(x)=5x+3
1. Horizontal Stretch: Multiply the x-values by 3 to change the coefficient of x from 5 to 15. This gives us f(x/3)=15(x/3)+3.
2. Vertical Translation: Subtract 15 from the y-values to change the constant term from 3 to -12. This gives us f(x/3)-12=15(x/3)+3-12.
Combining the steps, we have the transformed function g(x)=15x-12.

Combination 2:
Start with f(x)=5x+3
1. Horizontal Compression: Divide the x-values by 1/3 to change the coefficient of x from 5 to 15. This gives us f(3x)=15(3x)+3.
2. Vertical Translation: Subtract 15 from the y-values to change the constant term from 3 to -12. This gives us f(3x)-12=15(3x)+3-12.
Combining the steps, we have the transformed function g(x)=15x-12.

In both combinations, we first modify the x-values to stretch or compress the function horizontally by changing the coefficient of x. Then we modify the y-values to translate the function vertically by changing the constant term. These combinations of transformations result in the given transformation from f(x)=5x+3 to g(x)=15x-12.