Find the inverse function of the function f(x)=4x^((5)/(3)) on the domain x >= 0. f^(-1)(x)=((x)/(4))^((3)/(5)) f^(-1)(x)=(x^(-(3)/(5)))/(4) f^(-1)(x)=(x^((3)/(5)))/(4) f^(-1)(x)=((x)/(4))^(-(3)/(5))

Understand function & domain: Understand the function and its domain. The given function is f(x) = 4x^(5/3), 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 = 4x^(5/3). We will solve this equation for x.Isolate x term: Isolate the term containing x. To isolate x, we first divide both sides of the equation by 4. y/4 = x^(5/3)Raise to power: Raise both sides of the equation to the power of (3/5) to cancel the exponent on x. (y/4)^(3/5) = (x^(5/3))^(3/5)Simplify right side: Simplify the right side of the equation using the property of exponents (a^(m))^n = a^(m*n). (y/4)^(3/5) = x^((5/3)*(3/5)) (y/4)^(3/5) = x^(1) (y/4)^(3/5) = xWrite inverse function: Write the inverse function. The inverse function is f^(-1)(x) = (x/4)^(3/5).

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Find the inverse function of the function
f(x)=4x^((5)/(3)) on the domain
x >= 0.

f^(-1)(x)=((x)/(4))^((3)/(5))

f^(-1)(x)=(x^(-(3)/(5)))/(4)

f^(-1)(x)=(x^((3)/(5)))/(4)

f^(-1)(x)=((x)/(4))^(-(3)/(5))

Find the inverse function of the function $f(x)=4 x^{\frac{5}{3}}$ on the domain $x \geq 0$ . $\newline$ $f^{-1}(x)=\left(\frac{x}{4}\right)^{\frac{3}{5}}$ $\newline$ $f^{-1}(x)=\frac{x^{-\frac{3}{5}}}{4}$ $\newline$ $f^{-1}(x)=\frac{x^{\frac{3}{5}}}{4}$ $\newline$ $f^{-1}(x)=\left(\frac{x}{4}\right)^{-\frac{3}{5}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Find the inverse function of the function

f(x)=4 x^{\frac{5}{3}}

on the domain

x \geq 0

\newline

f^{-1}(x)=\left(\frac{x}{4}\right)^{\frac{3}{5}}

\newline

f^{-1}(x)=\frac{x^{-\frac{3}{5}}}{4}

\newline

f^{-1}(x)=\frac{x^{\frac{3}{5}}}{4}

\newline

f^{-1}(x)=\left(\frac{x}{4}\right)^{-\frac{3}{5}}

Understand function & domain: Understand the function and its domain. $\newline$ The given function is $f(x) = 4x^{(5/3)}$ , 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 $y = 4x^{(5/3)}$ . We will solve this equation for $x$ .
Isolate x term: Isolate the term containing $x$ . $\newline$ To isolate $x$ , we first divide both sides of the equation by $4$ . $\newline$ $\frac{y}{4} = x^{\frac{5}{3}}$
Raise to power: Raise both sides of the equation to the power of $(\frac{3}{5})$ to cancel the exponent on $x$ . $(\frac{y}{4})^{(\frac{3}{5})} = (x^{(\frac{5}{3})})^{(\frac{3}{5})}$
Simplify right side: Simplify the right side of the equation using the property of exponents $(a^{m})^{n} = a^{m*n}$ . $\newline$ $(\frac{y}{4})^{\frac{3}{5}} = x^{(\frac{5}{3})\cdot(\frac{3}{5})}$ $\newline$ $(\frac{y}{4})^{\frac{3}{5}} = x^{1}$ $\newline$ $(\frac{y}{4})^{\frac{3}{5}} = x$
Write inverse function: Write the inverse function. $\newline$ The inverse function is $f^{-1}(x) = \left(\frac{x}{4}\right)^{\frac{3}{5}}$ .