If $f(x)=\dfrac{x-3}{x-4}$, then for what value of $x$ is $f^{-1}(x)$ undefined?
We start by finding the inverse function of $f$. By definition, we know that $f(f^{-1}(x)) = x$, so $$\frac{f^{-1}(x)-3}{f^{-1}(x)-4} = x.$$We can solve this equation for $f^{-1}(x)$. First we multiply both sides by $f^{-1}(x)-4$: $$f^{-1}(x)-3 = x\cdot(f^{-1}(x)-4).$$Then we expand: $$f^{-1}(x)-3 = x\cdot f^{-1}(x)-4x.$$Then we rearrange so as to group all terms involving $f^{-1}(x)$ on the left side: $$f^{-1}(x)-x\cdot f^{-1}(x) = 3-4x.$$We can factor on the left side: $$f^{-1}(x)\cdot (1-x) = 3-4x.$$Finally, we divide both sides by $1-x$ to obtain our inverse function, $$f^{-1}(x) = \frac{3-4x}{1-x}.$$This function is defined for all $x$ except $\boxed{1}$.
1
Level 5
Algebra
Problem: If $f(x)=\dfrac{x-3}{x-4}$, then for what value of $x$ is $f^{-1}(x)$ undefined?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine when the inverse function \( f^{-1}(x) \) is undefined, we first need to find the expression for \( f^{-1}(x) \). The function \( f(x) = \frac{x-3}{x-4} \) is a rational function, and we need to find its inverse by solving for \( x \) in terms of \( y \).

1. Set \( y = f(x) \), which gives us the equation \( y = \frac{x-3}{x-4} \).
2. Rearranging this equation, we multiply both sides by \( x-4 \) to eliminate the fraction: \( y(x-4) = x-3 \).
3. Expanding the left side results in \( yx - 4y = x - 3 \).
4. Next, we isolate \( x \) terms on one side: \( yx - x = 4y - 3 \).
5. Factoring out \( x \) gives us \( x(y - 1) = 4y - 3 \).
6. Finally, we solve for \( x \): \( x = \frac{4y - 3}{y - 1} \).

Now we have the inverse function \( f^{-1}(x) = \frac{4x - 3}{x - 1} \). 

Next, we need to identify when this inverse function is undefined. A rational function is undefined when its denominator is zero. Therefore, we set the denominator of \( f^{-1}(x) \) to zero:

1. Set \( x - 1 = 0 \).
2. Solving this gives \( x = 1 \).

Thus, \( f^{-1}(x) \) is undefined when \( x = 1 \). This leads us to conclude that the value of \( x \) for which \( f^{-1}(x) \) is undefined is \( 1 \).<|END_OF_LATENT|> The final answer is $1$.