For the function f(x)=3x^((1)/(3)), find f^(-1)(x). f^(-1)(x)=3x^(3) f^(-1)(x)=((x)/(3))^((1)/(3)) f^(-1)(x)=(x^(3))/(3) f^(-1)(x)=((x)/(3))^(3)

Rewrite function with y: To find the inverse function, f^(-1)(x), we need to switch the roles of x and y in the original function and then solve for y. Let's start by rewriting the function with y instead of f(x): y = 3x^(1/3)Switch x and y: Now, switch x and y to find the inverse: x = 3y^(1/3)Isolate y: To solve for y, we need to isolate y on one side of the equation. Start by dividing both sides by 3: x/3 = y^(1/3)Eliminate cube root: Now, raise both sides of the equation to the power of 3 to eliminate the cube root: (x/3)^3 = (y^(1/3))^3Simplify and find inverse: Simplifying both sides gives us: (x/3)^3 = yWe have now isolated y and found the inverse function: f^(-1)(x) = (x/3)^3

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For the function
f(x)=3x^((1)/(3)), find
f^(-1)(x).

f^(-1)(x)=3x^(3)

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

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

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

For the function $f(x)=3 x^{\frac{1}{3}}$ , find $f^{-1}(x)$ . $\newline$ $f^{-1}(x)=3 x^{3}$ $\newline$ $f^{-1}(x)=\left(\frac{x}{3}\right)^{\frac{1}{3}}$ $\newline$ $f^{-1}(x)=\frac{x^{3}}{3}$ $\newline$ $f^{-1}(x)=\left(\frac{x}{3}\right)^{3}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

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

, find

f^{-1}(x)

\newline

f^{-1}(x)=3 x^{3}

\newline

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

\newline

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

\newline

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

Rewrite function with y: To find the inverse function, $f^{-1}(x)$ , we need to switch the roles of $x$ and $y$ in the original function and then solve for $y$ . Let's start by rewriting the function with $y$ instead of $f(x)$ : $\newline$ $y = 3x^{\frac{1}{3}}$
Switch x and y: Now, switch x and y to find the inverse: $x = 3y^{(1/3)}$
Isolate y: To solve for y, we need to isolate y on one side of the equation. Start by dividing both sides by $3$ : $\frac{x}{3} = y^{\frac{1}{3}}$
Eliminate cube root: Now, raise both sides of the equation to the power of $3$ to eliminate the cube root: $(\frac{x}{3})^3 = (y^{\frac{1}{3}})^3$
Simplify and find inverse: Simplifying both sides gives us: $\newline$ egin{equation}(\frac{x}{ $3$ })^ $3$ = y\end{equation}
Simplify and find inverse: Simplifying both sides gives us: $\newline$ $(\frac{x}{3})^3 = y$ We have now isolated $y$ and found the inverse function: $\newline$ $f^{-1}(x) = (\frac{x}{3})^3$