Find the inverse function of the function f(x)=2x^((3)/(5)) on the domain x >= 0. f^(-1)(x)=((x)/(2))^((5)/(3)) f^(-1)(x)=((x)/(2))^(-(5)/(3)) f^(-1)(x)=(x^(-(5)/(3)))/(2) f^(-1)(x)=(x^((5)/(3)))/(2)

Understand function and domain: Understand the function and its domain. The given function is f(x) = 2x^(3/5), and the domain is x >= 0. To find the inverse function, we need to solve for x in terms of y, where y = f(x).Replace with y: Replace f(x) with y to prepare for finding the inverse. Let y = 2x^(3/5). Our goal is to express x in terms of y.Isolate term with exponent: Isolate the term with the exponent. Divide both sides of the equation by 2 to isolate the x term. y/2 = x^(3/5)Raise to reciprocal: Raise both sides of the equation to the reciprocal of 3/5 to solve for x. (y/2)^(5/3) = (x^(3/5))^(5/3)Simplify right side: Simplify the right side of the equation. Since (a^(m))^n = a^(m*n), we have: (y/2)^(5/3) = x^((3/5)*(5/3)) (y/2)^(5/3) = x^(1) (y/2)^(5/3) = xWrite inverse function: Write the inverse function. The inverse function is f^(-1)(x) = (x/2)^(5/3).

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Find the inverse function of the function
f(x)=2x^((3)/(5)) on the domain
x >= 0.

f^(-1)(x)=((x)/(2))^((5)/(3))

f^(-1)(x)=((x)/(2))^(-(5)/(3))

f^(-1)(x)=(x^(-(5)/(3)))/(2)

f^(-1)(x)=(x^((5)/(3)))/(2)

Find the inverse function of the function $f(x)=2 x^{\frac{3}{5}}$ on the domain $x \geq 0$ . $\newline$ $f^{-1}(x)=\left(\frac{x}{2}\right)^{\frac{5}{3}}$ $\newline$ $f^{-1}(x)=\left(\frac{x}{2}\right)^{-\frac{5}{3}}$ $\newline$ $f^{-1}(x)=\frac{x^{-\frac{5}{3}}}{2}$ $\newline$ $f^{-1}(x)=\frac{x^{\frac{5}{3}}}{2}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Find the inverse function of the function

f(x)=2 x^{\frac{3}{5}}

on the domain

x \geq 0

\newline

f^{-1}(x)=\left(\frac{x}{2}\right)^{\frac{5}{3}}

\newline

f^{-1}(x)=\left(\frac{x}{2}\right)^{-\frac{5}{3}}

\newline

f^{-1}(x)=\frac{x^{-\frac{5}{3}}}{2}

\newline

f^{-1}(x)=\frac{x^{\frac{5}{3}}}{2}

Understand function and domain: Understand the function and its domain. $\newline$ The given function is $f(x) = 2x^{(3/5)}$ , and the domain is $x \geq 0$ . To find the inverse function, we need to solve for $x$ in terms of $y$ , where $y = f(x)$ .
Replace with $y$ : Replace $f(x)$ with $y$ to prepare for finding the inverse. $\newline$ Let $y = 2x^{(3/5)}$ . Our goal is to express $x$ in terms of $y$ .
Isolate term with exponent: Isolate the term with the exponent. $\newline$ Divide both sides of the equation by $2$ to isolate the $x$ term. $\newline$ $\frac{y}{2} = x^{\frac{3}{5}}$
Raise to reciprocal: Raise both sides of the equation to the reciprocal of $\frac{3}{5}$ to solve for $x$ . $\left(\frac{y}{2}\right)^{\frac{5}{3}} = \left(x^{\frac{3}{5}}\right)^{\frac{5}{3}}$
Simplify right side: Simplify the right side of the equation. $\newline$ Since $(a^{m})^{n} = a^{m*n}$ , we have: $\newline$ $(\frac{y}{2})^{\frac{5}{3}} = x^{(\frac{3}{5})*(\frac{5}{3})}$ $\newline$ $(\frac{y}{2})^{\frac{5}{3}} = x^{1}$ $\newline$ $(\frac{y}{2})^{\frac{5}{3}} = x$
Write inverse function: Write the inverse function. $\newline$ The inverse function is $f^{-1}(x) = \left(\frac{x}{2}\right)^{\frac{5}{3}}$ .