For the function f(x)=(root(7)(x)+5)/(4), find f^(-1)(x). f^(-1)(x)=4(x^(7)-5) f^(-1)(x)=4x^(7)-5 f^(-1)(x)=(4x-5)^(7) f^(-1)(x)=4(x-5)^(7)

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: f(x) = (root(7)(x) + 5) / 4 y = (x^(1/7) + 5) / 4Replace variables: Now, replace f(x) with y and x with f^(-1)(x): x = (y^(1/7) + 5) / 4Multiply by 4: Next, we need to solve for y. Multiply both sides by 4 to get rid of the denominator: 4x = y^(1/7) + 5Subtract 5: Subtract 5 from both sides to isolate the term with y: 4x - 5 = y^(1/7)Raise to power 7: Now, raise both sides to the power of 7 to get rid of the 7th root: (4x - 5)^7 = yWrite inverse function: Finally, we can write the inverse function by replacing y with f^(-1)(x): f^(-1)(x) = (4x - 5)^7

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For the function
f(x)=(root(7)(x)+5)/(4), find
f^(-1)(x).

f^(-1)(x)=4(x^(7)-5)

f^(-1)(x)=4x^(7)-5

f^(-1)(x)=(4x-5)^(7)

f^(-1)(x)=4(x-5)^(7)

For the function $f(x)=\frac{\sqrt[7]{x}+5}{4}$ , find $f^{-1}(x)$ . $\newline$ $f^{-1}(x)=4\left(x^{7}-5\right)$ $\newline$ $f^{-1}(x)=4 x^{7}-5$ $\newline$ $f^{-1}(x)=(4 x-5)^{7}$ $\newline$ $f^{-1}(x)=4(x-5)^{7}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

f(x)=\frac{\sqrt[7]{x}+5}{4}

, find

f^{-1}(x)

\newline

f^{-1}(x)=4\left(x^{7}-5\right)

\newline

f^{-1}(x)=4 x^{7}-5

\newline

f^{-1}(x)=(4 x-5)^{7}

\newline

f^{-1}(x)=4(x-5)^{7}

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 $y$ : $f(x) = \left(\sqrt[7]{x} + 5\right) / 4$ $y = \left(x^{1/7} + 5\right) / 4$
Replace variables: Now, replace $f(x)$ with $y$ and $x$ with $f^{-1}(x)$ : $x = \frac{y^{\frac{1}{7}} + 5}{4}$
Multiply by $4$ : Next, we need to solve for $y$ . Multiply both sides by $4$ to get rid of the denominator: $\newline$ $4x = y^{\frac{1}{7}} + 5$
Subtract $5$ : Subtract $5$ from both sides to isolate the term with $y$ : $4x - 5 = y^{1/7}$
Raise to power $7$ : Now, raise both sides to the power of $7$ to get rid of the $7$ th root: $\newline$ $(4x - 5)^7 = y$
Write inverse function: Finally, we can write the inverse function by replacing $y$ with $f^{-1}(x)$ : $f^{-1}(x) = (4x - 5)^7$