Domain Of A Function Practice Problems

The domain of a function is a fundamental concept in mathematics, representing the set of all possible input values (often denoted as x) for which the function produces a valid output. Understanding and determining the domain of a function is crucial in various mathematical contexts, from graphing and calculus to real-world applications. Mastering domain determination requires practice, and this article provides a comprehensive set of practice problems to solidify your understanding.

Understanding the Domain of a Function

Before diving into practice problems, let's revisit the core concept. The domain of a function f(x) is the set of all x values that can be plugged into the function without causing any undefined operations. These undefined operations generally include:

• Division by zero: A function is undefined if the denominator is zero.
• Taking the square root (or any even root) of a negative number: This results in a complex number, which is not typically considered within the realm of real-valued functions unless explicitly stated.
• Taking the logarithm of a non-positive number: Logarithms are only defined for positive arguments.

Therefore, when determining the domain, you need to identify any values of x that would lead to these undefined operations and exclude them from the domain.

Types of Functions and their Domain Considerations

Different types of functions have different considerations when determining their domains. Let's briefly discuss some common types:

• Polynomial Functions: These functions, like f(x) = x^2 + 3x - 5 or f(x) = 7x^5 - 2x + 1, generally have a domain of all real numbers, denoted as (-∞, ∞), because there are no restrictions on the values you can plug in.
• Rational Functions: These are functions that can be expressed as a ratio of two polynomials, such as f(x) = (x + 1) / (x - 2). The domain excludes any x values that make the denominator zero.
• Radical Functions: These functions involve radicals, such as square roots, cube roots, etc. For even roots (square root, fourth root, etc.), the expression inside the radical must be greater than or equal to zero. For odd roots (cube root, fifth root, etc.), there are no restrictions on the expression inside the radical, and the domain is all real numbers.
• Logarithmic Functions: These functions, like f(x) = ln(x) or f(x) = log(x), are only defined for positive arguments. The expression inside the logarithm must be strictly greater than zero.
• Trigonometric Functions: Functions like sine (sin(x)) and cosine (cos(x)) have a domain of all real numbers. Tangent (tan(x)) and secant (sec(x)) have restricted domains due to the presence of cosine in the denominator of their definitions. Cotangent (cot(x)) and cosecant (csc(x)) also have restricted domains due to sine in the denominator.

Practice Problems: Finding the Domain

Now, let's tackle a variety of practice problems to hone your skills in determining the domain of different types of functions. Each problem will be followed by a detailed solution and explanation.

Problem 1:

Find the domain of the function f(x) = 3x + 5.

Solution:

This is a polynomial function. There are no restrictions on the values of x that can be plugged into this function. Therefore, the domain is all real numbers.

Domain: (-∞, ∞)

Problem 2:

Find the domain of the function g(x) = (x + 2) / (x - 3).

Solution:

This is a rational function. The denominator cannot be equal to zero. Therefore, we need to find the value(s) of x that make the denominator zero:

x - 3 = 0 x = 3

Thus, x cannot be equal to 3. The domain is all real numbers except 3.

Domain: (-∞, 3) ∪ (3, ∞)

Problem 3:

Find the domain of the function h(x) = √(x - 4).

Solution:

This is a radical function with a square root (an even root). The expression inside the square root must be greater than or equal to zero:

x - 4 ≥ 0 x ≥ 4

Therefore, the domain is all real numbers greater than or equal to 4.

Domain: [4, ∞)

Problem 4:

Find the domain of the function k(x) = ln(x + 1).

Solution:

This is a logarithmic function. The argument of the logarithm must be strictly greater than zero:

x + 1 > 0 x > -1

Therefore, the domain is all real numbers greater than -1.

Domain: (-1, ∞)

Problem 5:

Find the domain of the function m(x) = 1 / √(2 - x).

Solution:

This function combines both a rational function and a radical function. The expression inside the square root must be greater than or equal to zero, and the denominator cannot be equal to zero. Therefore, the expression inside the square root must be strictly greater than zero:

2 - x > 0 2 > x x < 2

Therefore, the domain is all real numbers less than 2.

Domain: (-∞, 2)

Problem 6:

Find the domain of the function p(x) = ³√(x + 5).

Solution:

This is a radical function with a cube root (an odd root). There are no restrictions on the expression inside a cube root. Therefore, the domain is all real numbers.

Domain: (-∞, ∞)

Problem 7:

Find the domain of the function q(x) = (x - 1) / ((x + 2)(x - 4))

Solution:

This is a rational function. We need to find the values of x that make the denominator zero:

(x + 2)(x - 4) = 0 x + 2 = 0 or x - 4 = 0 x = -2 or x = 4

Therefore, x cannot be equal to -2 or 4. The domain is all real numbers except -2 and 4.

Domain: (-∞, -2) ∪ (-2, 4) ∪ (4, ∞)

Problem 8:

Find the domain of the function r(x) = √(x^2 - 9).

Solution:

This is a radical function with a square root. The expression inside the square root must be greater than or equal to zero:

x^2 - 9 ≥ 0 x^2 ≥ 9

This inequality is satisfied when x ≥ 3 or x ≤ -3.

Therefore, the domain is (-∞, -3] ∪ [3, ∞).

Problem 9:

Find the domain of the function s(x) = log((x - 2)/(x + 1))

Solution:

This is a logarithmic function with a rational expression as its argument. The argument of the logarithm must be strictly greater than zero:

(x - 2) / (x + 1) > 0

To solve this inequality, we consider the critical points where the numerator or denominator is zero: x = 2 and x = -1. We then test intervals to determine where the expression is positive:

• x < -1: Choose x = -2. ((-2) - 2) / ((-2) + 1) = (-4) / (-1) = 4 > 0. So, the interval (-∞, -1) is part of the solution.
• -1 < x < 2: Choose x = 0. (0 - 2) / (0 + 1) = -2 < 0. So, the interval (-1, 2) is not part of the solution.
• x > 2: Choose x = 3. (3 - 2) / (3 + 1) = 1 / 4 > 0. So, the interval (2, ∞) is part of the solution.

Therefore, the domain is (-∞, -1) ∪ (2, ∞).

Problem 10:

Find the domain of the function t(x) = √(4 - x^2)

Solution:

This is a radical function with a square root. The expression inside the square root must be greater than or equal to zero:

4 - x^2 ≥ 0 x^2 ≤ 4 -2 ≤ x ≤ 2

Therefore, the domain is [-2, 2].

Problem 11:

Find the domain of the function u(x) = e^(1/(x-1))

Solution:

This function involves an exponential function and a rational expression. The exponential function e^x is defined for all real numbers x. However, the exponent is a rational expression, 1/(x-1), which is undefined when x = 1. Therefore, the domain is all real numbers except x = 1.

Domain: (-∞, 1) ∪ (1, ∞)

Problem 12:

Find the domain of the function v(x) = arcsin(x/3)

Solution:

The arcsin function, also written as sin⁻¹(x), is the inverse sine function. Its domain is [-1, 1]. Therefore, for arcsin(x/3) to be defined:

-1 ≤ x/3 ≤ 1 -3 ≤ x ≤ 3

Therefore, the domain is [-3, 3].

Problem 13:

Find the domain of the function w(x) = (√(x+4))/(x-1)

Solution:

This function combines a square root and a rational function. The expression inside the square root must be greater than or equal to zero, and the denominator cannot be zero. This gives us two conditions:

• x + 4 ≥ 0 => x ≥ -4
• x - 1 ≠ 0 => x ≠ 1

Combining these conditions, the domain is all real numbers greater than or equal to -4, except for 1.

Domain: [-4, 1) ∪ (1, ∞)

Problem 14:

Find the domain of the function y(x) = √(x^2 + 1)

Solution:

This is a radical function with a square root. The expression inside the square root must be greater than or equal to zero:

x^2 + 1 ≥ 0

Since x^2 is always non-negative (greater than or equal to zero) for any real number x, adding 1 to it will always result in a positive number. Therefore, x^2 + 1 is always greater than zero for all real numbers x.

Domain: (-∞, ∞)

Problem 15:

Find the domain of the function z(x) = tan(x)

Solution:

The tangent function, tan(x), is defined as sin(x) / cos(x). The domain is restricted by the values of x for which cos(x) = 0. Cosine is zero at x = π/2 + nπ, where n is any integer. Therefore, the domain of tan(x) is all real numbers except x = π/2 + nπ.

Domain: x ≠ π/2 + nπ, where n is an integer. This can also be written as the union of open intervals: ... ∪ (-3π/2, -π/2) ∪ (-π/2, π/2) ∪ (π/2, 3π/2) ∪ ...

Problem 16:

Find the domain of the function f(x) = cot(x)

Solution:

The cotangent function, cot(x), is defined as cos(x) / sin(x). The domain is restricted by the values of x for which sin(x) = 0. Sine is zero at x = nπ, where n is any integer. Therefore, the domain of cot(x) is all real numbers except x = nπ.

Domain: x ≠ nπ, where n is an integer. This can also be written as the union of open intervals: ... ∪ (-π, 0) ∪ (0, π) ∪ (π, 2π) ∪ ...

Problem 17:

Find the domain of the function g(x) = csc(x)

Solution:

The cosecant function, csc(x), is defined as 1 / sin(x). The domain is restricted by the values of x for which sin(x) = 0. Sine is zero at x = nπ, where n is any integer. Therefore, the domain of csc(x) is all real numbers except x = nπ.

Domain: x ≠ nπ, where n is an integer. This is the same domain as cotangent.

Problem 18:

Find the domain of the function h(x) = sec(x)

Solution:

The secant function, sec(x), is defined as 1 / cos(x). The domain is restricted by the values of x for which cos(x) = 0. Cosine is zero at x = π/2 + nπ, where n is any integer. Therefore, the domain of sec(x) is all real numbers except x = π/2 + nπ.

Domain: x ≠ π/2 + nπ, where n is an integer. This is the same domain as tangent.

Problem 19:

Find the domain of the function k(x) = √(ln(x))

Solution:

This function combines a square root and a logarithm. We have two restrictions:

1. The argument of the logarithm must be positive: x > 0
2. The expression inside the square root must be non-negative: ln(x) ≥ 0

To solve ln(x) ≥ 0, we take the exponential of both sides:

e^(ln(x)) ≥ e^0 x ≥ 1

Combining x > 0 and x ≥ 1, we get x ≥ 1.

Domain: [1, ∞)

Problem 20:

Find the domain of the function m(x) = ln(√(x+2))

Solution:

This function combines a logarithm and a square root. We have two restrictions:

1. The expression inside the square root must be greater than or equal to zero: x + 2 ≥ 0 => x ≥ -2
2. The argument of the logarithm must be strictly greater than zero: √(x + 2) > 0

Squaring both sides of the second inequality:

x + 2 > 0 x > -2

Combining x ≥ -2 and x > -2, we get x > -2.

Domain: (-2, ∞)

Conclusion

Determining the domain of a function is a fundamental skill in mathematics. By understanding the restrictions imposed by different types of functions and practicing with a variety of problems, you can master this concept. Remember to always consider division by zero, even roots of negative numbers, and logarithms of non-positive numbers when determining the domain. The practice problems presented here offer a solid foundation for further exploration and application of this crucial mathematical concept. Continuously practice and apply these concepts to solidify your understanding and build confidence in your problem-solving abilities. Good luck!