For the function f(x)=4x^((1)/(5)), find f^(-1)(x). f^(-1)(x)=((x)/(4))^(5) f^(-1)(x)=(x^(5))/(4) f^(-1)(x)=(x^((1)/(5)))/(4) f^(-1)(x)=(4x)^(5)

Rewrite 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 = 4x^(1/5)Switch x and y: Now, switch x and y to find the inverse: x = 4y^(1/5)Isolate y: To solve for y, we need to isolate y on one side of the equation. Start by dividing both sides by 4: x/4 = y^(1/5)Raise to power of 5: Now, raise both sides of the equation to the power of 5 to eliminate the fifth root on the right side: (x/4)^5 = (y^(1/5))^5Simplify right side: Simplifying the right side, we get y by itself because (y^(1/5))^5 = y: (x/4)^5 = yInverse function: Now we have the inverse function: f^(-1)(x) = (x/4)^5

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For the function
f(x)=4x^((1)/(5)), find
f^(-1)(x).

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

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

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

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

For the function $f(x)=4 x^{\frac{1}{5}}$ , find $f^{-1}(x)$ . $\newline$ $f^{-1}(x)=\left(\frac{x}{4}\right)^{5}$ $\newline$ $f^{-1}(x)=\frac{x^{5}}{4}$ $\newline$ $f^{-1}(x)=\frac{x^{\frac{1}{5}}}{4}$ $\newline$ $f^{-1}(x)=(4 x)^{5}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

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

, find

f^{-1}(x)

\newline

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

\newline

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

\newline

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

\newline

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

Rewrite 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 $f(x)$ : $\newline$ $y = 4x^{\frac{1}{5}}$
Switch x and y: Now, switch x and y to find the inverse: $x = 4y^{1/5}$
Isolate y: To solve for y, we need to isolate y on one side of the equation. Start by dividing both sides by $4$ : $\newline$ $\frac{x}{4} = y^{\frac{1}{5}}$
Raise to power of $5$ : Now, raise both sides of the equation to the power of $5$ to eliminate the fifth root on the right side: $\newline$ (\frac{x}{$4$})^\(5 = (y^{\frac{ $1$ }{ $5$ }})^ $5$
Simplify right side: Simplifying the right side, we get $y$ by itself because $(y^{\frac{1}{5}})^5 = y$ : $\left(\frac{x}{4}\right)^5 = y$
Inverse function: Now we have the inverse function: $f^{-1}(x) = \left(\frac{x}{4}\right)^5$