For the function f(x)=((x)/(5))^((1)/(3)), find f^(-1)(x). f^(-1)(x)=(5x)^(3) f^(-1)(x)=5x^(3) f^(-1)(x)=((x)/(5))^(3) f^(-1)(x)=(5x)^((1)/(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 = ((x)/(5))^((1)/(3))Switch x and y: Now, we switch x and y to find the inverse: x = ((y)/(5))^((1)/(3))Isolate y: Next, we want to isolate y. To do this, we'll raise both sides of the equation to the power of 3 to get rid of the cube root: x^3 = (((y)/(5))^((1)/(3)))^3Raise both sides to power of 3: When we raise a power to a power, we multiply the exponents. Since (1/3) * 3 = 1, the cube root and the power of 3 cancel each other out, leaving us with: x^3 = y/5Multiply both sides by 5: Now, we multiply both sides by 5 to solve for y: 5 * x^3 = yFind inverse function: We have found the inverse function. The inverse function, f^(-1)(x), is: f^(-1)(x) = 5x^3

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For the function
f(x)=((x)/(5))^((1)/(3)), find
f^(-1)(x).

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

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

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

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

For the function $f(x)=\left(\frac{x}{5}\right)^{\frac{1}{3}}$ , find $f^{-1}(x)$ . $\newline$ $f^{-1}(x)=(5 x)^{3}$ $\newline$ $f^{-1}(x)=5 x^{3}$ $\newline$ $f^{-1}(x)=\left(\frac{x}{5}\right)^{3}$ $\newline$ $f^{-1}(x)=(5 x)^{\frac{1}{3}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

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

, find

f^{-1}(x)

\newline

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

\newline

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

\newline

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

\newline

f^{-1}(x)=(5 x)^{\frac{1}{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 $f(x)$ : $\newline$ $y = \left(\frac{x}{5}\right)^{\frac{1}{3}}$
Switch x and y: Now, we switch x and y to find the inverse: $x = \left(\frac{y}{5}\right)^{\frac{1}{3}}$
Isolate y: Next, we want to isolate y. To do this, we'll raise both sides of the equation to the power of $3$ to get rid of the cube root: $x^3 = \left(\frac{y}{5}\right)^{\left(\frac{1}{3}\right)^3}$
Raise both sides to power of $3$ : When we raise a power to a power, we multiply the exponents. Since $(\frac{1}{3}) \times 3 = 1$ , the cube root and the power of $3$ cancel each other out, leaving us with: $x^3 = \frac{y}{5}$
Multiply both sides by $5$ : Now, we multiply both sides by $5$ to solve for y: $\newline$ $5 \times x^3 = y$
Find inverse function: We have found the inverse function. The inverse function, $f^{-1}(x)$ , is: $f^{-1}(x) = 5x^3$