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The function [tex]$f(x) = 5^x$[/tex] is dilated to become [tex]$g(x) = 5^{\frac{1}{d} x}$[/tex]. What is the effect on [tex]f(x)[tex]$[/tex]?

A. [tex]$[/tex]f(x)$[/tex] is stretched horizontally by a factor of 4.
B. [tex]$f(x)$[/tex] is stretched vertically by a factor of 4.
C. [tex]$f(x)[tex]$[/tex] is compressed vertically by a factor of [tex]$[/tex]\frac{1}{4}$[/tex].
D. [tex]$f(x)$[/tex] is compressed horizontally by a factor of [tex]$\frac{1}{4}$[/tex].

Sagot :

GinnyAnswer
To determine the effect on the function [tex]\( f(x) = 5^x \)[/tex] when it is transformed into [tex]\( g(x) = 5^{\frac{x}{d}} \)[/tex], let's analyze the transformation and its implications carefully.

Given:
1. The original function is [tex]\( f(x) = 5^x \)[/tex].
2. The transformed function is [tex]\( g(x) = 5^{\frac{x}{d}} \)[/tex].

We need to figure out how this transformation changes the graph of [tex]\( f(x) \)[/tex].

### Step-by-Step Analysis:

1. Understanding the Transformation:
- The function [tex]\( g(x) = 5^{\frac{x}{d}} \)[/tex] means that for every [tex]\( x \)[/tex] in the domain of [tex]\( g(x) \)[/tex], it is scaled by the factor [tex]\( \frac{1}{d} \)[/tex].
- Essentially, the exponent [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] is replaced by [tex]\( \frac{x}{d} \)[/tex].

2. Horizontal Stretch/Compression:
- Replacing [tex]\( x \)[/tex] with [tex]\( \frac{x}{d} \)[/tex] in the exponent means that [tex]\( x \)[/tex] is divided by [tex]\( d \)[/tex] before being put into the function [tex]\( 5^x \)[/tex].
- This division by [tex]\( d \)[/tex] stretches the graph horizontally if [tex]\( d > 1 \)[/tex] and compresses it horizontally if [tex]\( 0 < d < 1 \)[/tex].

3. Effect of the Factor [tex]\( d \)[/tex]:
- If [tex]\( d = 4 \)[/tex], then [tex]\( g(x) = 5^{\frac{x}{4}} \)[/tex].
- This specific transformation [tex]\( \frac{x}{4} \)[/tex] means that each [tex]\( x \)[/tex] value in [tex]\( f(x) \)[/tex] becomes [tex]\( 4x \)[/tex] in [tex]\( g(x) \)[/tex], causing a horizontal stretch by a factor of [tex]\( 4 \)[/tex].

### Conclusion:
Given that [tex]\( f(x) \)[/tex] is stretched horizontally by a factor of 4, we can conclude that:

The correct effect on [tex]\( f(x) \)[/tex] is that it is stretched horizontally by a factor of 4.

So the correct answer is:

- [tex]\( f(x) \)[/tex] is stretched horizontally by a factor of 4.

This matches the observation given.
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