- The function composition \((f \circ g)(x)\) is found by evaluating \(f(g(x))\).
- Substitute \(g(x) = \frac{2}{x}\) into \(f(x) = x + 1\) to get \(f(g(x)) = \frac{2}{x} + 1\).
- Comparing this with the provided choices confirms that the correct expression is \(\frac{2}{x} + 1\).
- Final Answer: \(\boxed{\frac{2}{x} + 1}\)
Explanation
- Understanding the Problem and Function Composition
We are given two functions, \(f(x) = x + 1\) and \(g(x) = \frac{2}{x}\). We need to determine the expression equivalent to the composition \((f \circ g)(x)\), which is defined as \(f(g(x))\). The function composition means that we substitute the output of the function \(g(x)\) as the input for the function \(f(x)\).
- Computing the Composition of Functions
To find \(f(g(x))\), we take the expression for \(g(x)\), which is \(\frac{2}{x}\), and substitute it into the function \(f(x)\). Since \(f(x) = x + 1\), substituting \(g(x)\) for \(x\) gives:
\[f(g(x)) = g(x) + 1 = \frac{2}{x} + 1\]
- Evaluating the Options and Identifying the Correct Answer
We have the result \(f(g(x)) = \frac{2}{x} + 1\). Comparing this result with the provided options:
- \(\frac{2}{x + 1}\)
- \(\frac{2x + 2}{x}\)
- \(\frac{2}{x} + 1\)
- \(\frac{2}{x^2 + x}\)
The expression \(\frac{2}{x} + 1\) is directly listed as the third option.
Examples
Function composition is a fundamental concept in mathematics that allows us to chain processes together. For example, if you have a currency converter that changes dollars to euros (function \(g\)) and then a second process that adds a tax fee to the euro amount (function \(f\)), the combined process is represented by the composition \(f(g(x))\). Understanding how to simplify such expressions ensures you can correctly determine the net outcome of sequential operations in finance, engineering, or everyday planning.