To find the inverse of the function $g(x) = -\frac{1}{x-1} + 3$, we will follow these standard steps:
Step 1: Replace $g(x)$ with $y$.
$$y = -\frac{1}{x-1} + 3$$
Step 2: Swap $x$ and $y$.
To find the inverse, we switch the input and output variables.
$$x = -\frac{1}{y-1} + 3$$
Step 3: Solve for $y$.
We need to isolate $y$ on one side of the equation.
First, subtract 3 from both sides to move the constant term:
$$x - 3 = -\frac{1}{y-1}$$
Next, multiply both sides by $(y - 1)$ to get it out of the denominator:
$$(x - 3)(y - 1) = -1$$
Now, divide both sides by $(x - 3)$ to isolate the $(y - 1)$ term:
$$y - 1 = \frac{-1}{x - 3}$$
Finally, add 1 to both sides to solve for $y$:
$$y = \frac{-1}{x - 3} + 1$$
We can simplify the fraction $\frac{-1}{x - 3}$ by moving the negative sign to the denominator (since $\frac{-1}{a} = \frac{1}{-a}$):
$$\frac{-1}{x - 3} = \frac{1}{-(x - 3)} = \frac{1}{3 - x}$$
So, the equation becomes:
$$y = \frac{1}{3 - x} + 1$$
Step 4: Replace $y$ with the inverse notation $g^{-1}(x)$.
$$g^{-1}(x) = \frac{1}{3 - x} + 1$$
*(Note: The form $g^{-1}(x) = -\frac{1}{x - 3} + 1$ is also correct, but simplifying the signs is usually preferred.)*
Final Answer:
g^{-1}(x) = \frac{1}{3 - x} + 1
Parent Tip: Review the logic above to help your child master the concept of inverse function worksheet college algebra.
