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A function is considered one-to-one if each output value is uniquely paired with one input value. This means that different inputs should result in different outputs. It's like having a unique fingerprint for each input. To check if a function is one-to-one, we can use the horizontal line test: if a horizontal line intersects the graph of the function more than once, it's not one-to-one. In our example, the function given is \( f(x) = \frac{1}{x} \). We assumed \( f(a) = f(b) \), leading us to \( \frac{1}{a} = \frac{1}{b} \). By cross-multiplying, we conclude that \( a = b \). This shows that any two distinct inputs yield distinct outputs, confirming that \( f(x) = \frac{1}{x} \) is indeed one-to-one.

• A one-to-one function ensures each output is paired with only one input.
• The horizontal line test is a helpful tool for identification.
• In our case, \( \frac{1}{a} = \frac{1}{b} \) implies \( a = b \), verifying the function is one-to-one.

The domain of a function defines all the possible input values that the function can accept. It's essential to understand the domain when dealing with functions because it tells us the limits within which the function operates. For \( f(x) = \frac{1}{x} \), we look for values of \( x \) for which the function is defined. Given that division by zero is undefined, \( x = 0 \) is excluded from the domain. Thus, the domain of \( f(x) = \frac{1}{x} \) is all real numbers except zero, which is mathematically expressed as \( x \in \mathbb{R} \setminus \{0\} \). Recognizing this exclusion is crucial for both the function and its inverse.

• The domain is the set of all inputs a function can accept.
• For \( f(x) = \frac{1}{x} \), the input cannot be zero.
• Thus, the domain is all real numbers except zero: \( \mathbb{R} \setminus \{0\} \).

Finding the inverse of a function involves reversing the roles of inputs and outputs. If our function maps \( x \) to \( y \), its inverse will map \( y \) back to \( x \). To find an inverse, we typically solve for the original input variable in terms of the output. For \( f(x) = \frac{1}{x} \), we set it equal to \( y \), giving us \( y = \frac{1}{x} \). Solving for \( x \), by multiplying both sides by \( x \), we find \( xy = 1 \). We then isolate \( x \) as \( x = \frac{1}{y} \). With this, the inverse function, denoted as \( f^{-1}(x) \), is \( \frac{1}{x} \). It's important to verify that both \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \), ensuring that the inverse is correct within the function's domain, which excludes zero.

• To find a function's inverse, swap inputs and outputs.
• For \( f(x) = \frac{1}{x} \), solve \( y = \frac{1}{x} \) for \( x \) to get the inverse.
• The inverse function is \( f^{-1}(x) = \frac{1}{x} \), ensuring valid mappings back to original inputs.