Problem 33 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\sqrt{x}, g(x)=x+3$$

Step-by-Step Solution

Answer

a. \(f \circ g = \sqrt{x + 3}\) b. The domain of \(f \circ g\) is \([-3, \infty)\)

Step 1: Compute \(f \circ g\)

Begin by substituting \(g(x)\) into \(f(x)\) to compute \(f \circ g\). That is, wherever you see an \(x\) in \(f\), replace it with the entire expression for \(g(x)\). In other words, \(f \circ g = f(g(x)) = \sqrt{g(x)} = \sqrt{x + 3}\). So, \(f \circ g = \sqrt{x + 3}\).

Step 2: Determine the domain of \(f \circ g\)

Now determine which values of \(x\) can be safely used as input to \(f \circ g\). Keep in mind that the square root function \(f(x)\) is real if and only if \(x \geq 0\). Thus, the expression under the square root in \(f \circ g\), which is \(x + 3\), must be nonnegative: \(x + 3 \geq 0\). Solving for \(x\), finds that \(x \geq -3\). Therefore, the domain of \(f \circ g\) is \([-3, \infty)\).

Key Concepts

Square Root Function

The square root function is quite unique as it is represented by the expression \(f(x) = \sqrt{x}\). This function takes a non-negative number as input and returns a non-negative number as output. Essentially, the function outputs the number that, when multiplied by itself, gives the original input number. For example, if \(x = 4\), \( \sqrt{4} = 2\), since \(2 \times 2 = 4\).

Understanding how the square root function operates is essential for utilizing it correctly in mathematical problems. It emphasizes how input restrictions apply to ensure valid outputs. For instance:

• The input must be a non-negative number, meaning \(x \geq 0\)
• The output is also non-negative, referred to as the principal square root

These properties are crucial when performing operations like function composition where the restrictions need to be considered. The square root must be handled carefully to maintain valid mathematical operations throughout problem-solving.

Domain of a Function

The domain of a function refers to all the possible input values, or "x-values," that can be used without breaking any mathematical rules. Every function has its own specific domain, dictated by its formula. Understanding the domain is crucial for applying functions correctly in equations and real-world scenarios.

Considerations for determining the domain often include:

• Ensuring there are no zero denominators in fractions
• Keeping the expression inside a square root non-negative, for real numbers
• Avoiding logarithms of non-positive numbers

In the case of the square root function, \(f(x) = \sqrt{x}\), since the input \(x\) must be non-negative, the domain of \(f(x)\) is \([0, \infty)\). This emphasizes the concept that the domain is determined by ensuring the function's conditions are satisfied for real numbers. Choosing the domain carefully is necessary for accurate mathematical reasoning.

Domain Restrictions

Domain restrictions are conditions or rules applied to the input values of functions to ensure the function is defined and operates correctly. In composition of functions, like \((f \circ g)(x)\), these restrictions become even more important as they help in identifying valid input ranges that satisfy both the input and composition functions.

For one, functions often need specific restrictions such as:

• Square roots require non-negative numbers inside
• Denominators in fractions must not be zero
• Functions involving logarithms need positive arguments

In the given exercise, the composition \(f(g(x)) = \sqrt{x + 3}\) has a restriction that \(x + 3\) must be non-negative for the square root to remain valid. Solving this gives a domain of \(x \geq -3\), hence the domain of \(f \circ g\) is \([-3, \infty)\). Properly applying domain restrictions guarantees the function remains meaningful and computationally accurate.