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Understanding how to find the inverse of a function is crucial in calculus and algebra. To invert a function, you simply reverse the roles of the inputs and outputs. For the exercise given, we were asked to find the inverse of the function:

f(x) = 7x - 1

We start by rewriting the function in a more familiar form by replacing the function notation, \(f(x)\), with \(y\). This gives us:

y = 7x - 1

Next, to find the inverse, we swap \(x\) and \(y\). This action represents the essence of finding an inverse: switching the dependent and independent variables. Our equation turns into:

x = 7y - 1

Now, our task is to solve for \(y\), but remember, this \(y\) will be our inverted function's value. Finally, we will replace \(y\) with \(f^{-1}(x)\) to express the inverse function properly.

Solving equations forms the backbone of finding function inverses. Once we swap \(x\) and \(y\), we start our journey to isolate \(y\). Let's break down the steps:

• Add 1 to both sides:

x + 1 = 7y

By doing this, we successfully start isolating our \(y\). The idea here is to make \(y\) the subject of the formula. It鈥檚 crucial to treat both sides equally.

• Next, divide both sides by 7:

y = \frac{x + 1}{7}

This step gives us the formula of our inverse function. It鈥檚 important to perform the operations in sequence: First, addition/subtraction, then multiplication/division. This logical approach will always guide us to the correct inverse.

Algebraic manipulation is another critical concept required for function inversion. Manipulating equations correctly allows you to isolate desired variables. In this exercise, we start with the equation:

x = 7y - 1

First, we isolated the term involving \(y\) by adding 1 to both sides:

x + 1 = 7y

We used addition here to eliminate the constant term. Next, we handled the coefficient of \(y\) by dividing both sides by 7:

y = \frac{x + 1}{7}

This ensures that \(y\) is isolated and the equation represents our inverted function. Finally, we exchanged \(y\) with \(f^{-1}(x)\):

f^{-1}(x) = \frac{x + 1}{7}

These steps of adding, subtracting, multiplying, and dividing are what make algebraic manipulation such a powerful tool in problem-solving in mathematics.