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In mathematics, the domain and range of a function are key concepts that help us understand the behavior of functions. The **domain** of a function represents all possible inputs (usually represented by "x") that the function can accept. In easier terms, it's every number you can plug into the function.

For example, if a problem states that the domain of a function is \((-fty, fty)\), it means you can use any real number. There are no restrictions.

Meanwhile, the **range** of a function includes all possible outputs (usually represented by "f(x)") that the function might produce after being applied to its domain. Understanding this relationship is fundamental when dealing with inverse functions because it helps predict and manipulate inputs and outputs.

If a function \( f(x) \) and its inverse \( f^{-1}(x) \) have the same domain and range of \((-fty, fty)\), they are defined everywhere for every real number, which simplifies the process of solving equations involving these functions.

Solving functions involves determining the value of a variable that satisfies an equation. This is crucial when working with inverse functions because understanding the forward and inverse operations aids in isolating and determining unknowns.

In the exercise, to solve \(7 + f^{-1}(x-1) = 9\), it means we first isolate the inverse function by subtracting 7 from both sides, resulting in \(f^{-1}(x-1) = 2\). At this point, understanding the relationship between \(f\) and \(f^{-1}\) becomes essential.

• Recall that if \(f(a) = b\), then the inverse function \(f^{-1}(b) = a\).
• This property allows us to find \(x\) by using information given, such as \(f(2) = 6\), which implies that \(f^{-1}(6) = 2\).

Thus, when we equate \(x-1 = 6\), by adding 1 to both sides, \(x = 7\).In the real world, this process can be applied to many scenarios, such as solving for time or distance in physics problems.

Equation manipulation is a fundamental skill in solving mathematical problems. It involves using arithmetic operations to isolate variables and simplify expressions, which is particularly useful when working with functions and their inverses.

For instance, in part (b) of the exercise, we start with the equation \(4 + f(x+3) = -3\). The first step is to simplify by subtracting 4 from both sides, leading to \(f(x+3) = -7\). This is a classic example of isolating the term that contains the function.

Next, apply the inverse function property: if \(f^{-1}(-7) = 0\), this means \(f(0) = -7\). By recognizing this relationship, you equate \(x+3 = 0\).

By subtracting 3 from both sides of the equation, the solution is \(x = -3\).

• Each operation has the aim of progressively reducing the complexity of the problem.
• It’s a step-by-step method to reach a solution.

By mastering equation manipulation, you can handle more complex functions and equations, increasing your problem-solving abilities significantly.