For the function f(x)=(-6)/(5-4x), find f^(-1)(x). Answer: f^(-1)(x)=

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 = (-6)/(5 - 4x)Interchange x and y: Now, interchange x and y to find the inverse: x = (-6)/(5 - 4y)Multiply both sides: Next, we solve for y. Start by multiplying both sides of the equation by (5 - 4y) to get rid of the fraction: x(5 - 4y) = -6Distribute x: Distribute x on the left side of the equation: 5x - 4xy = -6Isolate terms with y: Now, we want to isolate terms with y on one side. Let's add 4xy to both sides: 5x = 4xy - 6Add 6 to both sides: Next, add 6 to both sides to isolate the terms with y on the right side: 5x + 6 = 4xyFactor out y: Now, we need to factor out y on the right side: 5x + 6 = y(4x)Divide both sides: Finally, divide both sides by 4x to solve for y: y = (5x + 6) / (4x) This is the inverse function, f^(-1)(x).

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For the function
f(x)=(-6)/(5-4x), find
f^(-1)(x).
Answer:
f^(-1)(x)=

For the function $f(x)=\frac{-6}{5-4 x}$ , find $f^{-1}(x)$ . $\newline$ Answer: $f^{-1}(x)=$

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Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

f(x)=\frac{-6}{5-4 x}

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

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 $f(x)$ : $\newline$ $y = \frac{-6}{5 - 4x}$
Interchange $x$ and $y$ : Now, interchange $x$ and $y$ to find the inverse: $\newline$ $x = \frac{-6}{5 - 4y}$
Multiply both sides: Next, we solve for $y$ . Start by multiplying both sides of the equation by $(5 - 4y)$ to get rid of the fraction: $\newline$ $x(5 - 4y) = -6$
Distribute $x$ : Distribute $x$ on the left side of the equation: $5x - 4xy = -6$
Isolate terms with y: Now, we want to isolate terms with y on one side. Let's add $4xy$ to both sides: $\newline$ $5x = 4xy - 6$
Add $6$ to both sides: Next, add $6$ to both sides to isolate the terms with $y$ on the right side: $\newline$ $5x + 6 = 4xy$
Factor out y: Now, we need to factor out y on the right side: $\newline$ $5x + 6 = y(4x)$
Divide both sides: Finally, divide both sides by $4x$ to solve for $y$ : $\newline$ $y = \frac{5x + 6}{4x}$ $\newline$ This is the inverse function, $f^{-1}(x)$ .