To determine the effect on the function [tex]\( f(x) = 5^x \)[/tex] when it is transformed to [tex]\( g(x) = 3 \cdot 5^x \)[/tex], let's consider the changes made to the function.
For the given function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 5^x \][/tex]
The transformation that is applied to [tex]\( f(x) \)[/tex] is a multiplication by 3:
[tex]\[ g(x) = 3 \cdot 5^x \][/tex]
To understand the effect of this transformation, we need to recognize what multiplying a function by a factor outside of the function does.
1. Vertical Stretch/Compression: When a function [tex]\( f(x) \)[/tex] is multiplied by a constant [tex]\( a \)[/tex] to form [tex]\( g(x) = a \cdot f(x) \)[/tex], the effect is a vertical dilation (stretch or compression) by the factor [tex]\( a \)[/tex].
- If [tex]\( a > 1 \)[/tex], the function is stretched vertically.
- If [tex]\( 0 < a < 1 \)[/tex], the function is compressed vertically.
- If [tex]\( a \)[/tex] is negative, there is also a reflection over the x-axis, but in this case, [tex]\( a = 3 \)[/tex] which is positive.
Since in our transformation, [tex]\( g(x) = 3 \cdot 5^x \)[/tex], the constant 3 indicates a vertical stretch:
[tex]\[ g(x) = 3 \cdot f(x) \][/tex]
Thus, the function [tex]\( f(x) = 5^x \)[/tex] is transformed by a vertical stretch. The factor of this vertical stretch is 3.
So, the correct statement is:
[tex]\[ f(x) \text{ is stretched vertically by a factor of 3.} \][/tex]
The other options do not apply:
- [tex]\( f(x) \)[/tex] is not stretched horizontally by a factor of 3.
- [tex]\( f(x) \)[/tex] is not compressed horizontally by a factor of [tex]\(\frac{1}{3}\)[/tex].
- [tex]\( f(x) \)[/tex] is not compressed vertically by a factor of [tex]\(\frac{1}{3}\)[/tex].
Therefore, the correct answer is:
[tex]\[ f(x) \text{ is stretched vertically by a factor of 3}. \][/tex]
