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

Write function as y: To find the inverse function, we first write the function as y = ((x-5)/7)^(1/5).Swap x and y: Next, we swap x and y to get x = ((y-5)/7)^(1/5).Eliminate fifth root: Now we solve for y. Raise both sides of the equation to the power of 5 to eliminate the fifth root: x^5 = (y-5)/7.Isolate term with y: Multiply both sides by 7 to isolate the term with y: 7x^5 = y - 5.Solve for y: Add 5 to both sides to solve for y: 7x^5 + 5 = y.Inverse function: Now we have the inverse function: f^(-1)(x) = 7x^5 + 5.

Home

Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. For the function

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

, find

f^{-1}(x)

\newline

f^{-1}(x)=7 x^{5}+5

\newline

f^{-1}(x)=(7 x+5)^{5}

\newline

f^{-1}(x)=7(x+5)^{5}

\newline

f^{-1}(x)=7\left(x^{5}+5\right)

Write function as $y$ : To find the inverse function, we first write the function as $y = \left(\frac{x-5}{7}\right)^{\frac{1}{5}}$ .
Swap x and y: Next, we swap x and y to get $x = \left(\frac{y-5}{7}\right)^{\frac{1}{5}}$ .
Eliminate fifth root: Now we solve for $y$ . Raise both sides of the equation to the power of $5$ to eliminate the fifth root: $x^5 = (y-5)/7$ .
Isolate term with y: Multiply both sides by $7$ to isolate the term with $y$ : $7x^5 = y - 5$ .
Solve for y: Add $5$ to both sides to solve for y: $7x^5 + 5 = y$ .
Inverse function: Now we have the inverse function: $f^{-1}(x) = 7x^5 + 5$ .