Find the inverse \( f(x)=5 x+7 \)

To find the inverse of the function \( f(x) = 5x + 7 \), start by replacing \( f(x) \) with \( y \): \( y = 5x + 7 \). Next, switch \( x \) and \( y \) to express the function as an equation for \( y \): \( x = 5y + 7 \). Now, solve for \( y \) by isolating it: \( x - 7 = 5y \) then divide both sides by 5: \( y = \frac{x - 7}{5} \). Therefore, the inverse function is: \( f^{-1}(x) = \frac{x - 7}{5} \). Interestingly, finding inverses is not just math wizardry; it has real-world applications! For example, in finance, if you have an investment formula where future value influences current investment, knowing how to reverse the process helps you determine your current needs to reach a future goal. Flexing those inverse muscles can really come in handy! Many people overlook that their functions are indeed one-to-one (where each output corresponds to only one input), which is crucial for inverses to exist. A common mistake is failing to check this before proceeding. In our example, since the function is linear and has a non-zero slope, it's one-to-one! Always confirm that the function is invertible before diving into the calculations.