Question

Which best describes how the graph of g(x) = square²⁵/9x relates to the graph of the parent function, f(x)=sqtˣ ?

Answer 1

The graph of
`\(g(x) = \frac{\sqrt[25]{9x}}{√(x)}\)` is a compressed and vertically stretched version of the parent function
`\(f(x) = √(x)\)`.

Explanation:

The given function
`\(g(x) = \frac{\sqrt[25]{9x}}{√(x)}\)` can be analyzed in terms of its relationship with the parent function
`\(f(x) = √(x)\)`. The parent function f(x) represents the square root of x, and any transformation of this function involves modifications to its amplitude, compression, or stretch.

In the given expression
`\(g(x) = \frac{\sqrt[25]{9x}}{√(x)}\)`, the presence of
`\(\sqrt[25]{9x}\)`suggests a 25th root applied to 9x, which introduces a horizontal compression due to the fractional exponent. Additionally, the sqrt{x} in the denominator implies a vertical stretch. These combined transformations result in a graph that is compressed horizontally and stretched vertically compared to the parent function f(x).

To illustrate, consider the effect of the 25th root on the x-values, making the graph narrower. Simultaneously, the square root in the denominator stretches the y-values. These transformations collectively yield a graph that maintains the essential shape of the square root function but appears compressed horizontally and stretched vertically, differentiating it from the original parent function.

Answer 2

The graph of
`\( g(x) = \left((√(25))/(9)x\right)^2 \)` is a vertically compressed and vertically stretched version of the parent function
`\( f(x) = √(x) \)`.

Step-by-step explanation:

The function g(x) represents a transformation of the parent function f(x). The expression
`\( \left((√(25))/(9)x\right)^2 \)` indicates a vertical compression by a factor of
`\( (1)/(3) \)` and a vertical stretch by a factor of 3 applied to
`\( f(x) = √(x) \)`.

The square inside the function affects the vertical dimension, making the graph wider or narrower compared to the original function. The fraction
`\( (1)/(3) \)` in front of x indicates a compression, while the fraction 3 squared implies a stretch.

Understanding the impact of different transformations on functions is essential in graphing and analyzing mathematical expressions. It enables a comprehensive exploration of how changes in the function's structure affect its graph, providing a foundation for advanced mathematical concepts and applications.