Problem 46 - Calculus Volume 1

Question

For each pair of functions, find a. \((f \circ g)(x)\) and b. \((g \circ f)(x)\) Simplify the results. Find the domain of each of the results. \(\quad f(x)=\sqrt{x}, g(x)=x+9\)

Step-by-Step Solution

Answer

a. \((f \circ g)(x) = \sqrt{x+9}\), Domain: \([-9, \infty)\); b. \((g \circ f)(x) = \sqrt{x} + 9\), Domain: \([0, \infty)\).

Step 1: Understanding Composition of Functions

The composition of functions involves applying one function to the results of another. For two functions \( f(x) \) and \( g(x) \), the notation \((f \circ g)(x)\) means you apply \(g\) first and then apply \(f\) to the result. Similarly, \((g \circ f)(x)\) means to apply \(f\) first and then \(g\).

Step 2: Calculate \((f \circ g)(x)\)

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). Since \(f(x) = \sqrt{x}\) and \(g(x) = x+9\), we have:\[(f \circ g)(x) = f(g(x)) = f(x+9) = \sqrt{x+9}\].

Step 3: Simplify \((f \circ g)(x)\) and Determine Its Domain

The expression \((f \circ g)(x) = \sqrt{x+9}\) is already simplified. The domain of this function is determined by the condition that the expression inside the square root must be non-negative, i.e., \(x+9 \geq 0\). Solving this inequality, we get \(x \geq -9\). Thus, the domain of \((f \circ g)(x)\) is \([-9, \infty)\).

Step 4: Calculate \((g \circ f)(x)\)

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). Since \(g(x) = x+9\) and \(f(x) = \sqrt{x}\), we have:\[(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 9\].

Step 5: Simplify \((g \circ f)(x)\) and Determine Its Domain

The expression \((g \circ f)(x) = \sqrt{x} + 9\) is already simplified. The domain of this function is determined by the condition that the expression inside the square root must be non-negative, i.e., \(x \geq 0\). Therefore, the domain of \((g \circ f)(x)\) is \([0, \infty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when we deal with different types of functions. The domain of a function is essentially the set of all possible input values (usually represented as 'x') that will produce a valid output when plugged into the function.
For most functions, you can consider the domain as all real numbers unless restrictions like division by zero or negative values under a square root are present.

• The domain for polynomial functions is typically all real numbers.
• However, for rational functions, the domain excludes values that make the denominator zero.
• For square root functions, the domain excludes values that result in a negative number under the root.

For example, if we take the function \(f(x) = \sqrt{x+9}\), the domain is derived from ensuring the expression \(x+9\) remains non-negative. Solving \(x+9 \geq 0\) gives \(x \geq -9\), making the domain \([-9, \infty)\). This means any value of \x\ that is -9 or larger will maintain a valid outcome.

Square Root Function

A square root function is a type of radical function and involves an expression under a root sign. The most basic form of this function is written as \(f(x) = \sqrt{x}\).
Square roots can be tricky because they only output non-negative results — this is why their domains must ensure that the input is non-negative as well. In other words, whatever is under the square root must be zero or more.

• Negative numbers aren't naturally in the domain because you can't take the square root of a negative number and still remain within the real numbers.
• The square root of zero is zero—this is the smallest result from a square root function.
• As the value inside the square root increases, so does the output of the function.

In function composition, such as \(f(x) = \sqrt{x+9}\), we analyze the square root's argument, \(x+9\). It must always be \(\geq0\), meaning \(x\geq-9\). The form and restrictions of square root functions make them distinct among functions.

Function Notation

Function notation is a convenient way to express functions and their compositions, such as \(f(x)\) or \(g(x)\).
This notation offers a clearer representation when defining multiple functions or their compositions, like \(f \circ g(x)\), which implies substituting \(g(x)\) as the input into \(f(x)\).

• \(f \circ g(x)\) is read as "f of g of x," meaning you perform \(g(x)\) first, then use that result as the input for \(f(x)\).
• Similarly, \(g \circ f(x)\) means applying \(f(x)\) first, then the function \(g\) to that result.

Function notation streamlines the process of applying functions sequentially, making it simpler to handle and understand complex operations. For instance, if \(f(x) = \sqrt{x}\) and \(g(x) = x+9\), \(f \circ g(x) = \sqrt{x+9}\) ensures you're applying the functions correctly. Understanding this notation helps in breaking down each step of composition, resulting in a clearer grasp of what the composite function represents.