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

Replace 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 replacing f(x) with y: y = (-3)/(1+3x)Switch x and y: Now, switch x and y to find the inverse: x = (-3)/(1+3y)Multiply by (1+3y): Next, we need to solve for y. Start by multiplying both sides of the equation by (1+3y) to get rid of the fraction: x * (1+3y) = -3Distribute x: Distribute x on the left side of the equation: x + 3xy = -3Isolate terms with y: Now, isolate terms with y on one side: 3xy = -3 - xFactor out y: Factor out y from the left side: y(3x) = -3 - xDivide by 3x: Divide both sides by 3x to solve for y: y = (-3 - x) / (3x)Split the fraction: Simplify the expression for y by splitting the fraction: y = -1/x - 3/(3x)Simplify the expression: Simplify the second term of the expression: y = -1/x - 1/xCombine the terms: Combine the terms: y = -2/xWrite the inverse function: Now that we have solved for y, we can write the inverse function: f^(-1)(x) = -2/x

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

For the function $f(x)=\frac{-3}{1+3 x}$ , find $f^{-1}(x)$ . $\newline$ Answer: $f^{-1}(x)=$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. For the function

f(x)=\frac{-3}{1+3 x}

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

Replace 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 replacing $f(x)$ with $y$ :
$y = \frac{-3}{1+3x}$
Switch x and y: Now, switch x and y to find the inverse: $x = \frac{-3}{1+3y}$
Multiply by $(1+3y)$ : Next, we need to solve for $y$ . Start by multiplying both sides of the equation by $(1+3y)$ to get rid of the fraction: $\newline$ $x \cdot (1+3y) = -3$
Distribute $x$ : Distribute $x$ on the left side of the equation: $\newline$ $x + 3xy = -3$
Isolate terms with y: Now, isolate terms with y on one side: $\newline$ $3xy = -3 - x$
Factor out $y$ : Factor out $y$ from the left side: $\newline$ $y(3x) = -3 - x$
Divide by $3x$ : Divide both sides by $3x$ to solve for $y$ : $y = \frac{-3 - x}{3x}$
Split the fraction: Simplify the expression for $y$ by splitting the fraction: $y = -\frac{1}{x} - \frac{3}{3x}$
Simplify the expression: Simplify the second term of the expression: $y = -\frac{1}{x} - \frac{1}{x}$
Combine the terms: Combine the terms: $y = -\frac{2}{x}$
Write the inverse function: Now that we have solved for $y$ , we can write the inverse function: $f^{-1}(x) = -\frac{2}{x}$