Domain of Functions

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What is the domain of a function?

The domain of a function is the set of all possible input values (usually x-values) that the function can accept.

How do you determine the domain of a simple polynomial function?

For a simple polynomial function, the domain is all real numbers because polynomials are defined for all x-values.

What is the domain of the function f(x) = 1/x?

The domain of f(x) = 1/x is all real numbers except x ≠ 0, since division by zero is undefined.

What changes in the domain for a function f(x) = √x?

The domain for f(x) = √x is all x ≥ 0, since the square root of a negative number is not defined in real numbers.

How do you find the domain of a rational function like f(x) = (x+1)/(x-2)?

To find the domain of a rational function, set the denominator not equal to zero. For f(x) = (x+1)/(x-2), the domain is all x ≠ 2.

Explain how logarithms affect the domain of a function.

The domain of a logarithmic function like f(x) = log(x) is x > 0, as logarithms are undefined for zero and negative numbers.

If a function f(x) is defined by f(x) = 1/(x^2 - 4), what is its domain?

For f(x) = 1/(x^2 - 4), find the values that make the denominator zero: x^2 - 4 = 0, x = ±2. Thus, the domain is all x ≠ ±2.

What is the implication of having a square root in the denominator of a function?

If there is a square root in the denominator, the expression inside the square root must be greater than zero.

How do you determine the domain of a piecewise function?

Identify the domain for each piece separately, based on their individual expressions and conditions, then combine them.

For the function f(x) = √(x-3), what restriction does the square root impose on the domain?

The domain of f(x) = √(x-3) is x ≥ 3, because the expression inside the square root must be non-negative.

How does an absolute value affect the domain of a function?

An absolute value function typically does not impose domain restrictions unless paired with other operations that do.

If a function has a fraction with a variable in the denominator, what must you exclude from the domain?

You must exclude values that make the denominator zero, as division by zero is undefined.

For a composite function like h(x) = sqrt(x - 5)/(x + 3), how do you find the domain?

For h(x) = sqrt(x - 5)/(x + 3), determine where x - 5 ≥ 0 (x ≥ 5) and x + 3 ≠ 0 (x ≠ -3). The domain is x ≥ 5, x ≠ -3.

What strategy can be used to determine the domain of a function with multiple operations?

Identify domain restrictions for each operation separately (e.g., no zero denominator, no negatives under even roots) and combine them.

Can the domain of a function ever be written in interval notation?

Yes, the domain can be expressed in interval notation indicating the range of x-values that are acceptable for the function.




Test Your Knowledge

Select the correct option


1. What is the domain of a function?

The domain is the range of output values (y-values) the function can produce.

The domain of a function is the set of all possible input values (usually x-values) that the function can accept.

The domain includes all possible values excluding those that make the function non-continuous.

The domain refers to the set of situations where the slope of the function is positive.

2. What is the domain of the function f(x) = 1/x?

The domain of f(x) = 1/x is all real numbers except x ≠ 0, since division by zero is undefined.

The domain of f(x) = 1/x is all real numbers except x > 1.

The domain of f(x) = 1/x is all positive real numbers.

The domain of f(x) = 1/x is all negative real numbers.

3. Explain how logarithms affect the domain of a function.

Logarithms expand the domain to include all numbers except integers.

Logarithms have no effect on the domain of a function.

The domain of a logarithmic function like f(x) = log(x) is x ≥ 0 and includes zero.

The domain of a logarithmic function like f(x) = log(x) is x > 0, as logarithms are undefined for zero and negative numbers.

4. How do you determine the domain of a simple polynomial function?

The domain is limited only to non-negative integers.

For a simple polynomial function, the domain is all real numbers because polynomials are defined for all x-values.

Simple polynomials have a domain restricted to even numbers only.

The domain is all positive integers.

5. If a function f(x) is defined by f(x) = 1/(x^2 - 4), what is its domain?

The domain is all x < 2.

The domain is all real numbers except x ≠ 4.

The domain is all real numbers.

For f(x) = 1/(x^2 - 4), find the values that make the denominator zero: x^2 - 4 = 0, x = ±2. Thus, the domain is all x ≠ ±2.