Suppose we’re given the following function

$$ \begin{align*} f(x) = \frac{8x - 7}{7x + 4}. \end{align*} $$

and we want to find its inverse along with its domain and range.

Finding The Inverse

To find the inverse, we just need to following simple steps. First replace \(f(x)\) with \(y\).

$$ \begin{align*} y = \frac{8x - 7}{7x + 4}. \end{align*} $$

Next, swap \(y\) with \(x\).

$$ \begin{align*} x = \frac{8y - 7}{7y + 4}. \end{align*} $$

Now, let’s solve for \(y\)

$$ \begin{align*} x &= \frac{8y - 7}{7y + 4} \\ x(7y + 4) &= 8y - 7 \\ 7xy + 4x &= 8y - 7 \\ 8y - 7xy &= 4x + 7 \\ y(8 - 7x) &= 4x + 7 \\ y &= \frac{4x + 7}{8 - 7x}. \\ \end{align*} $$

To verify our work is correct, we can take the composition of \(f\) and \(f^{-1}\) and we should expect to get \(x\). Let’s test this,

$$ \begin{align*} f^{-1}(f(x)) &= \frac{4\big( \frac{8x - 7}{7x + 4} \big) + 7}{8 - 7\big( \frac{8x - 7}{7x + 4} \big)} \\ &= \frac{\frac{4(8x - 7)}{7x + 4} + \frac{7(7x + 4)}{7x + 4}}{8 - 7\big( \frac{8x - 7}{7x + 4} \big)} \\ &= \frac{\frac{4(8x - 7)}{7x + 4} + \frac{7(7x + 4)}{7x + 4}}{\frac{8(7x + 4)}{7x + 4} - \frac{7(8x - 7)}{7x + 4}} \\ &= \frac{\frac{4(8x - 7)}{7x + 4} + \frac{7(7x + 4)}{7x + 4}}{\frac{8(7x + 4)}{7x + 4} - \frac{7(8x - 7)}{7x + 4}} \\ &= \frac{4(8x - 7) + 7(7x + 4)}{8(7x + 4) - 7(8x - 7)} \\ &= \frac{32x - 28 + 49x + 28}{56x + 32 - 56x + 49} \\ &= \frac{81x}{81} \\ & = x. \end{align*} $$

Which is what we wanted! We can also try the other direction but let’s just not do it now.

The Domain

The domain of the inverse of \(f\) is simply all real numbers except for anything that makes the denominator 0. Recall that the inverse function was,

$$ \begin{align*} f^{-1}(x) = \frac{4x + 7}{8 - 7x}. \end{align*} $$

The denominator is zero when \(8 - 7x = 0\). This means this happens when \(x = \frac{8}{7}\). So in other words, the domain is \((-\infty, \frac{8}{7}) \cup (\frac{8}{7}, \infty)\).

The Range

The range of the inverse of \(f\) is simply the range of our original function \(f\). Recall that the function was,

$$ \begin{align*} f(x) = \frac{8x - 7}{7x + 4}. \end{align*} $$

The denominator is zero when \(7x + 4 = 0\). This means this happens when \(x = \frac{-4}{7}\). So in other words the range is \((-\infty, \frac{-4}{7}) \cup (\frac{-4}{7}, \infty)\).
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