Domain Of A Function Practice Problems

Mastering the Domain of a Function: Practice Problems and Solutions

Understanding the domain of a function is fundamental to mastering precalculus and calculus. The domain represents all possible input values (x-values) for which the function is defined. A function is considered undefined when its output yields an undefined mathematical operation, such as division by zero or taking the square root of a negative number. This article provides a comprehensive exploration of domain problems, ranging from simple polynomial functions to more complex rational, radical, and logarithmic functions. We’ll cover numerous practice problems with detailed solutions to solidify your understanding.

Understanding the Concept of Domain

Before diving into practice problems, let's briefly revisit the core concept. The domain of a function is the set of all possible input values (typically x) for which the function is defined and produces a real output. Conversely, the range represents all possible output values (typically y). Identifying the domain is crucial for accurate function analysis, graphing, and solving equations.

Identifying Potential Issues:

Several scenarios can lead to a function being undefined:

• Division by Zero: A function containing a fraction will be undefined when the denominator is equal to zero. You must find the values of x that make the denominator zero and exclude them from the domain.

• Even Roots of Negative Numbers: Functions involving square roots, fourth roots, or any even root are undefined when the radicand (the expression inside the root) is negative. You need to ensure the radicand is greater than or equal to zero.

• Logarithms of Non-Positive Numbers: Logarithmic functions (like log_bx) are only defined for positive input values. The argument of the logarithm (the expression inside the logarithm) must be greater than zero.

Practice Problems: A Gradual Progression

We'll progress through problems of increasing complexity, starting with simpler polynomial functions and then tackling rational, radical, and logarithmic functions. Each problem will include a detailed solution, highlighting the key steps involved in determining the domain.

Polynomial Functions:

Problem 1: Find the domain of the function f(x) = 3x² + 2x - 5.

Solution: Polynomial functions are defined for all real numbers. There are no restrictions on the input x. Therefore, the domain of f(x) is (-∞, ∞) or all real numbers.

Problem 2: Find the domain of g(x) = x³ - 7x + 10.

Solution: Similar to Problem 1, this is a polynomial function. There are no restrictions on the input x, so the domain is (-∞, ∞) or all real numbers.

Rational Functions:

Problem 3: Find the domain of h(x) = (x + 2) / (x - 3).

Solution: Rational functions are fractions where the numerator and denominator are polynomials. The function is undefined when the denominator equals zero. We set the denominator equal to zero and solve for x:

x - 3 = 0 x = 3

The function is undefined when x = 3. Therefore, the domain is (-∞, 3) U (3, ∞). This means all real numbers except 3.

Problem 4: Find the domain of k(x) = (x² - 4) / (x² - 9x + 20).

Solution: Again, we need to find the values of x that make the denominator zero. Factor the denominator:

x² - 9x + 20 = (x - 4)(x - 5)

Setting the denominator equal to zero gives:

(x - 4)(x - 5) = 0 x = 4 or x = 5

The domain is all real numbers except 4 and 5. In interval notation: (-∞, 4) U (4, 5) U (5, ∞).

Radical Functions:

Problem 5: Find the domain of p(x) = √(x - 4).

Solution: The square root function is only defined for non-negative values. Therefore, the expression inside the square root must be greater than or equal to zero:

x - 4 ≥ 0 x ≥ 4

The domain is [4, ∞).

Problem 6: Find the domain of q(x) = √(9 - x²).

Solution: The radicand must be greater than or equal to zero:

9 - x² ≥ 0 x² ≤ 9 -3 ≤ x ≤ 3

The domain is [-3, 3].

Problem 7: Find the domain of r(x) = √(x) / (x - 1).

Solution: This combines a radical and a rational function. We have two conditions:

1. The radicand must be non-negative: x ≥ 0
2. The denominator cannot be zero: x ≠ 1

Combining these conditions, the domain is [0, 1) U (1, ∞).

Logarithmic Functions:

Problem 8: Find the domain of s(x) = log₂(x + 5).

Solution: Logarithmic functions are only defined for positive arguments. Therefore:

x + 5 > 0 x > -5

The domain is (-5, ∞).

Problem 9: Find the domain of t(x) = ln(x² - 1). (ln denotes the natural logarithm, base e)

Solution: The argument must be greater than zero:

x² - 1 > 0 x² > 1 x > 1 or x < -1

The domain is (-∞, -1) U (1, ∞).

Combining Function Types:

Problem 10: Find the domain of u(x) = √(x/(x-2)).

Solution: This problem combines a radical and a rational function.

1. The radicand must be non-negative: x/(x-2) ≥ 0. We analyze the sign of this expression using a sign chart. The critical points are x = 0 and x = 2.

   • If x < 0, both numerator and denominator are negative, resulting in a positive expression.
   • If 0 < x < 2, the numerator is positive and the denominator is negative, resulting in a negative expression.
   • If x > 2, both numerator and denominator are positive, resulting in a positive expression.

2. The denominator of the inner rational function cannot be zero: x ≠ 2

Combining these conditions, the domain is (-∞, 0] U (2, ∞)

Problem 11: Find the domain of v(x) = log₃(√(x+1) / (x-4)).

Solution:

1. Argument of the logarithm must be positive: √(x+1) / (x-4) > 0. The square root ensures the numerator is non-negative. Therefore, we focus on when the denominator is positive: x > 4. Also, the numerator cannot be zero, so x > -1.

2. The radicand must be non-negative: x + 1 ≥ 0 => x ≥ -1

3. The denominator cannot be zero: x ≠ 4

Combining these, we get the domain: (4, ∞)

Advanced Practice and Further Exploration

These problems provide a solid foundation for understanding domain restrictions. For further practice, consider exploring piecewise functions, trigonometric functions (where domains are restricted due to periodicity and undefined values), and inverse trigonometric functions. Remember to always analyze the components of the function separately to determine the overall domain.

By consistently practicing these types of problems, you will develop a strong intuition for identifying domain restrictions in various function types. This skill is crucial for success in higher-level mathematics. Remember to always check your work and visualize the function’s graph to ensure your understanding.