Question

it is useful to note that for most toolkit functions, a horizontal stretch or vertical stretch can be represented in other ways. for example, a horizontal compression of the function $f(x)=x^{2}$ by $1 / 2$ would result in a new function $g(x)=(2 x)^{2}$, but this can also be written as $g(x)=4 x^{2}$, a vertical stretch of $f(x)$ by 4. when writing a formula for a transformed toolkit, we only need to find one transformation that would produce the graph. ? exercise 1.5.7 write a formula for the toolkit square root function $f(x)=sqrt{x}$ horizontally stretched by three. $g(x)=$ question help: message instructor post to forum

Explanation:

Step1: Recall horizontal stretch formula

For a function \(y = f(x)\), a horizontal stretch by a factor \(a\) gives \(y = f(\frac{x}{a})\).

Step2: Apply to square - root function

Given \(f(x)=\sqrt{x}\) and \(a = 3\) (horizontal stretch by three), then \(g(x)=f(\frac{x}{3})\).
Substitute \(x\) with \(\frac{x}{3}\) in \(f(x)\): \(g(x)=\sqrt{\frac{x}{3}}\).

Answer:

\(g(x)=\sqrt{\frac{x}{3}}\)