Math Snap

STEP 1

1. We are given functions for which we need to find the inverse.
2. We need to show all work and use proper notation.
3. The process involves algebraic manipulation to express $y$ in terms of $x$.

STEP 2

1. Find the inverse of $f(x) = \frac{x-3}{x+2}$.
2. Find the inverse of $h(x) = \frac{4x-1}{2x+3}$.
3. Find the inverse of $y = \frac{x+4}{x-5}$.
4. Find the inverse of $g(x) = \frac{2x+1}{3x+7}$.

STEP 3

Given $f(x) = \frac{x-3}{x+2}$, we want to find its inverse.
1. Replace $f(x)$ with $y$:
$$y = \frac{x-3}{x+2}$$ 2. Swap $x$ and $y$:
$$x = \frac{y-3}{y+2}$$ 3. Solve for $y$:
$$x(y+2) = y-3$$ $$xy + 2x = y - 3$$ $$xy - y = -3 - 2x$$ $$y(x - 1) = -3 - 2x$$ $$y = \frac{-3 - 2x}{x - 1}$$ Thus, the inverse function is:
$$f^{-1}(x) = \frac{-3 - 2x}{x - 1}$$

STEP 4

Given $h(x) = \frac{4x-1}{2x+3}$, we want to find its inverse.
1. Replace $h(x)$ with $y$:
$$y = \frac{4x-1}{2x+3}$$ 2. Swap $x$ and $y$:
$$x = \frac{4y-1}{2y+3}$$ 3. Solve for $y$:
$$x(2y+3) = 4y - 1$$ $$2xy + 3x = 4y - 1$$ $$2xy - 4y = -3x - 1$$ $$y(2x - 4) = -3x - 1$$ $$y = \frac{-3x - 1}{2x - 4}$$ Thus, the inverse function is:
$$h^{-1}(x) = \frac{-3x - 1}{2x - 4}$$

STEP 5

Given $y = \frac{x+4}{x-5}$, we want to find its inverse.
1. Swap $x$ and $y$:
$$x = \frac{y+4}{y-5}$$ 2. Solve for $y$:
$$x(y-5) = y + 4$$ $$xy - 5x = y + 4$$ $$xy - y = 5x + 4$$ $$y(x - 1) = 5x + 4$$ $$y = \frac{5x + 4}{x - 1}$$ Thus, the inverse function is:
$$y^{-1}(x) = \frac{5x + 4}{x - 1}$$

Given $g(x) = \frac{2x+1}{3x+7}$, we want to find its inverse.
1. Replace $g(x)$ with $y$:
$$y = \frac{2x+1}{3x+7}$$ 2. Swap $x$ and $y$:
$$x = \frac{2y+1}{3y+7}$$ 3. Solve for $y$:
$$x(3y+7) = 2y + 1$$ $$3xy + 7x = 2y + 1$$ $$3xy - 2y = 1 - 7x$$ $$y(3x - 2) = 1 - 7x$$ $$y = \frac{1 - 7x}{3x - 2}$$ Thus, the inverse function is:
$$g^{-1}(x) = \frac{1 - 7x}{3x - 2}$$ The inverse functions are:
1. $f^{-1}(x) = \frac{-3 - 2x}{x - 1}$
2. $h^{-1}(x) = \frac{-3x - 1}{2x - 4}$
3. $y^{-1}(x) = \frac{5x + 4}{x - 1}$
4. $g^{-1}(x) = \frac{1 - 7x}{3x - 2}$