For the function f(x)=((x^((1)/(7)))/(5))^(3), find f^(-1)(x). f^(-1)(x)=(root(3)(5x))^(7) f^(-1)(x)=5root(3)(x^(7)) f^(-1)(x)=root(3)(5x^(7)) f^(-1)(x)=(5root(3)(x))^(7)

Replace and Switch: 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. The original function is f(x) = ((x^(1/7))/5)^3. Let's replace f(x) with y for clarity: y = ((x^(1/7))/5)^3 Now, switch x and y: x = ((y^(1/7))/5)^3 Next, we need to solve this equation for y.Cube Root Isolation: To isolate the term with y, we need to get rid of the cube on the right side. We can do this by taking the cube root of both sides: root(3)(x) = (y^(1/7))/5 Now we have the cube root of x equals y to the power of 1/7 divided by 5.Multiply by 5: To further isolate y, we need to multiply both sides by 5: 5 * root(3)(x) = y^(1/7) Now we have 5 times the cube root of x equals y to the power of 1/7.Raise to Power of 7: To solve for y, we need to raise both sides to the power of 7 to get rid of the exponent 1/7 on the right side: (5 * root(3)(x))^7 = y Now we have y on one side, and the expression (5 times the cube root of x) raised to the power of 7 on the other side.Final Inverse Function: We have found the inverse function: f^(-1)(x) = (5 * root(3)(x))^7 This is the inverse function of the original function f(x).

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For the function
f(x)=((x^((1)/(7)))/(5))^(3), find
f^(-1)(x).

f^(-1)(x)=(root(3)(5x))^(7)

f^(-1)(x)=5root(3)(x^(7))

f^(-1)(x)=root(3)(5x^(7))

f^(-1)(x)=(5root(3)(x))^(7)

For the function $f(x)=\left(\frac{x^{\frac{1}{7}}}{5}\right)^{3}$ , find $f^{-1}(x)$ . $\newline$ $f^{-1}(x)=(\sqrt[3]{5 x})^{7}$ $\newline$ $f^{-1}(x)=5 \sqrt[3]{x^{7}}$ $\newline$ $f^{-1}(x)=\sqrt[3]{5 x^{7}}$ $\newline$ $f^{-1}(x)=(5 \sqrt[3]{x})^{7}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

f(x)=\left(\frac{x^{\frac{1}{7}}}{5}\right)^{3}

, find

f^{-1}(x)

\newline

f^{-1}(x)=(\sqrt[3]{5 x})^{7}

\newline

f^{-1}(x)=5 \sqrt[3]{x^{7}}

\newline

f^{-1}(x)=\sqrt[3]{5 x^{7}}

\newline

f^{-1}(x)=(5 \sqrt[3]{x})^{7}

Replace and Switch: 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 $y$ . The original function is f(x) = igg(rac{x^{1/7}}{5}igg)^3. Let's replace $f(x)$ with $y$ for clarity: $\newline$ y = igg(rac{x^{1/7}}{5}igg)^3 $\newline$ Now, switch $x$ and $y$ : $\newline$ $x$ $0$ $\newline$ Next, we need to solve this equation for $y$ .
Cube Root Isolation: To isolate the term with $y$ , we need to get rid of the cube on the right side. We can do this by taking the cube root of both sides: $\newline$ $\sqrt[3]{x} = \frac{y^{\frac{1}{7}}}{5}$ $\newline$ Now we have the cube root of $x$ equals $y$ to the power of $\frac{1}{7}$ divided by $5$ .
Multiply by $5$ : To further isolate $y$ , we need to multiply both sides by $5$ : $5 \times \sqrt[3]{x} = y^{\frac{1}{7}}$ Now we have $5$ times the cube root of $x$ equals $y$ to the power of $\frac{1}{7}$ .
Raise to Power of $7$ : To solve for $y$ , we need to raise both sides to the power of $7$ to get rid of the exponent $\frac{1}{7}$ on the right side: $\newline$ $(5 \cdot \sqrt[3]{x})^7 = y$ $\newline$ Now we have $y$ on one side, and the expression ( $5$ times the cube root of $x$ ) raised to the power of $7$ on the other side.
Final Inverse Function: We have found the inverse function: $\newline$ $f^{-1}(x) = (5 \cdot \sqrt[3]{x})^7$ $\newline$ This is the inverse function of the original function $f(x)$ .