Problem 35 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=x^{2}+4, g(x)=\sqrt{1-x}$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 5 - x\) and the domain of \(f \circ g\) is \(x \leq 1\).

Step 1: Compute the Composite Function

The composite function \(f \circ g\) means to place the function \(g(x)\) inside the function \(f(x)\): \(f(g(x))\). The function \(g(x)\) is given as \(\sqrt{1-x}\) and the function \(f(x)\) uses its input squared and then adds 4. So, the composite function \(f(g(x))\) is computed as follows: \(f(g(x)) = f(\sqrt{1-x}) = (\sqrt{1-x})^{2} + 4 = 1 - x + 4 = 5 - x.\)

Step 2: Find the Domain of the Composite Function

Next, we will find the domain of the function \(f \circ g\). The domain of a function consists of all numbers \(x\) for which the function is defined, and the function \(f(g(x))\) is defined as long as the expression inside the square root in \(g(x)\) remains non-negative. That is, the domain is all \(x\) for which \(1 - x \geq 0\), or \(x \leq 1\). Combining this with the domain of \(f(x)\), which is all real numbers because function \(f\) involves just squaring and adding 4, a calculation that can be performed for any real number, we can conclude that the domain of the composite function \(f \circ g\) is all real x, such that \(x \leq 1\).

Key Concepts

Function Composition

Function composition is a process where one function is applied inside another function. This is denoted as

• \((f \circ g)(x)\) which represents function \(g(x)\) being applied first, and then its result being put into function \(f(x)\).
• In the exercise provided, \(g(x) = \sqrt{1-x}\) and \(f(x) = x^2 + 4\). So the function \(g(x)\) should be evaluated first and the result used in \(f(x)\).

This is why the composite function is also written as \(f(g(x))\). To solve such problems, follow these steps:

• Evaluate \(g(x)\): in our problem, this means applying the square root to \(1-x\).
• Then substitute \(g(x)\) into \(f(x)\): input the result from the first function into the second, replacing \(x\) in \(f(x)\) with \(g(x)\).
• Our solution reveals that \(f(g(x)) = 5 - x\).

This process simplifies complex problems into manageable steps. It is frequently used in calculus and algebra to combine multiple operations.

Domain of a Function

Understanding the domain of a function is crucial because it specifies the set of all possible input values (or \(x\) values) for which the function is defined. The domain of a function depends heavily on the operations involved in that function.

• For example, the domain for a function that involves a square root, like \(g(x) = \sqrt{1-x}\), is limited by the necessity to keep the expression inside the root non-negative.

Let's apply these ideas to find the domain of the composite function:

• Start by determining the domain for \(g(x)\), which requires \(1 - x \geq 0\), or \(x \leq 1\).
• Since \(f(x)\) contains operations that are defined for all real numbers, we focus on the restrictions imposed by \(g(x)\).
• Thus, the domain of the composite function \(f \circ g\), is all real numbers \(x\) such that \(x \leq 1\).

Understanding how to find the domain helps in ensuring the function's outputs are meaningful and valid.

Square Root Function

Square root functions are widely used in algebra, representing the operation of finding a number which, when multiplied by itself, gives the original number.

• The function \(g(x) = \sqrt{1-x}\) is an example of a square root function. Here, the operation involves finding the square root of the expression inside the root.
• A key aspect of working with square root functions is ensuring the expression under the root remains non-negative; otherwise, you would end up with an imaginary number, which is not included in the real number domain typically considered in such problems.

The behavior of square root functions impacts both the domain and the range:

• The domain is limited to values that make the expression under the root non-negative. In this exercise, making sure that \(1-x \geq 0\) led us to the domain \(x \leq 1\).
• Understanding these limitations is crucial when finding the domain of related composite functions.

Square root functions appear frequently and understanding their properties is fundamental for problem-solving across mathematics.