Find Each Of The Following Functions And State Their Domains

Find Each of the Following Functionsand State Their Domains

Understanding how to identify a function and determine its domain is a fundamental skill in algebra, calculus, and many applied fields. A function describes a relationship where each input (usually x) is assigned exactly one output (usually y or f(x)). The domain is the set of all permissible inputs for which the function produces a real‑valued output. Below is a step‑by‑step guide, illustrated with varied examples, that shows how to find each of the following functions and state their domains while avoiding common pitfalls.

Introduction: Why the Domain Matters

When you are asked to find each of the following functions and state their domains, the task has two parts:

1. Identify the rule that defines the function (often given as an algebraic expression, a piecewise description, or a graph).
2. Determine the domain—the collection of x‑values that keep the expression mathematically valid (no division by zero, no even‑root of a negative number, no logarithm of a non‑positive argument, etc.).

Mastering this process enables you to graph functions accurately, solve equations, and model real‑world phenomena without encountering undefined behavior.

Step‑by‑Step Procedure

1. Write the Function in Its Simplest Form

• Combine like terms, factor numerators and denominators, and cancel common factors only after noting any restrictions that arise from the original expression.
• Example: ( f(x)=\frac{x^2-4}{x-2} ) simplifies to ( f(x)=x+2 ) for ( x\neq2 ). The simplification hides the restriction ( x\neq2 ), which must be retained in the domain.

2. Identify Potential Restrictions Look for the following “red flags” in the function’s expression:

3. Solve the Inequalities or Equations

• For denominators, set the denominator equal to zero and solve.
• For radicals, set the radicand ≥ 0 and solve.
• For logarithms, set the argument > 0 and solve.
• Use factoring, the quadratic formula, or sign charts as needed.

4. Express the Domain

• Use interval notation (e.g., ((-\infty, -2)\cup(-2,3)\cup(3,\infty))) or set‑builder notation (e.g., ({x\in\mathbb{R}\mid x\neq -2, x\neq3})).
• If the function is defined for all real numbers, state the domain as ((-\infty,\infty)) or (\mathbb{R}).

5. Double‑Check for Hidden Restrictions

• After simplification, verify that any cancelled factor did not remove a necessary restriction.
• For composite functions (e.g., ( f(g(x)) )), ensure the inner function’s output lies within the domain of the outer function.

Worked Examples

Below are several typical functions. For each, we find the function (confirm its rule) and state its domain.

Example 1: Rational Function

[ f(x)=\frac{3x+5}{x^2-9} ]

Step 1: The rule is already given.
Step 2: Denominator (x^2-9\neq0).
Step 3: Solve (x^2-9=0\Rightarrow (x-3)(x+3)=0\Rightarrow x=3) or (x=-3).
Step 4: Exclude these points.

Domain: ((-\infty,-3)\cup(-3,3)\cup(3,\infty))

Note: No simplification removes the restriction, so the domain stands as is.

Example 2: Square‑Root Function

[ g(x)=\sqrt{2x-7} ]

Step 1: Rule: principal square root of (2x-7).
Step 2: Radicand must be non‑negative: (2x-7\ge0).
Step 3: Solve: (2x\ge7\Rightarrow x\ge\frac{7}{2}).

Domain: (\left[\frac{7}{2},\infty\right))

Example 3: Logarithmic Function

[ h(x)=\ln!\left(\frac{x+1}{x-4}\right) ]

Step 1: Rule: natural log of a rational expression.
Step 2: Argument must be positive: (\frac{x+1}{x-4}>0).
Step 3: Solve the inequality using a sign chart. Critical points at (x=-1) and (x=4).

We need the intervals where the expression is positive, excluding the zeros themselves because log(0) is undefined.

Domain: ((-\infty,-1)\cup(4,\infty))

Example 4

Worked Examples (Continued)

Below are several typical functions. For each, we find the function (confirm its rule) and state its domain.

Example 4: Composite Function

[ k(x) = \sqrt[3]{x+2} - 1 ]

Step 1: The rule is already given. Step 2: The cube root is defined for all real numbers. Step 3: There are no restrictions on the input.

Domain: ((-\infty,\infty))

Example 5: Function with a Piecewise Definition

[ l(x) = \begin{cases} x+1, & \text{if } x < 0 \ x^2, & \text{if } x \ge 0 \end{cases} ]

Step 1: Rule: The function is defined by two different expressions depending on the value of x. Step 2: For the first case, x < 0. There are no restrictions here. Step 3: For the second case, x ≥ 0. There are no restrictions here.

Domain: ((-\infty,\infty))

Example 6: Rational Function with a Restricted Denominator

[ m(x) = \frac{x-2}{x^2-4x+3} ]

Step 1: The rule is already given. Step 2: Denominator (x^2-4x+3\neq0). Step 3: Solve (x^2-4x+3=0\Rightarrow (x-1)(x-3)=0\Rightarrow x=1) or (x=3). Step 4: Exclude these points.

Domain: ((-\infty,1)\cup(1,3)\cup(3,\infty))

Note: No simplification removes the restriction, so the domain stands as is.

Conclusion

This guide provides a systematic approach to determining the domain of various functions. By carefully considering potential restrictions imposed by denominators, radicals, logarithms, and composite functions, we can accurately identify the set of all permissible input values. Remember to always double-check for hidden restrictions after simplification and to accurately represent the domain using either interval notation or set-builder notation. Mastering this process is crucial for successful problem-solving in algebra and calculus, ensuring that functions are applied correctly and avoiding undefined results. Practice with a variety of examples will solidify your understanding and build confidence in your ability to determine function domains.