Q: How to find domain of rational functions?

For rational functions f(x) = p(x)/q(x), the domain excludes values where the denominator equals zero. Set q(x) = 0 and solve for x. These x-values are excluded from the domain (vertical asymptotes). The domain is all real numbers except these values, written in interval notation with union symbols.

Q: How to find domain of radical functions?

For square root functions √(expression), the radicand (expression under the radical) must be non-negative. Set expression ≥ 0 and solve for x. This gives the domain. For even roots (√, ∜), require non-negative radicand. For odd roots (∛, ∛), all real numbers are allowed.

Q: What is the domain and range of logarithmic functions?

For logarithmic functions f(x) = log(x) or ln(x), the domain is x > 0 (positive real numbers only), written as (0, +∞). The range is all real numbers (-∞, +∞) because logarithms can output any real value. For log(g(x)), set g(x) > 0 to find the domain.

Question 1

What is domain and range in mathematics?

Accepted Answer

Domain is the set of all possible input values (x-values) for which a function is defined. Range is the set of all possible output values (y-values) that the function can produce. Together, they describe the complete behavior and limitations of a function.

Question 2

How do you find the domain and range of a function?

Accepted Answer

To find domain: identify values that make the function undefined (division by zero, negative under square root, log of non-positive). To find range: analyze function behavior, find minimum/maximum values, consider asymptotes, or graph the function. Use algebraic methods, graphing, or calculus techniques depending on the function type.

Question 3

What is interval notation for domain and range?

Accepted Answer

Interval notation uses brackets and parentheses to describe sets: [a, b] means a ≤ x ≤ b (closed interval, includes endpoints), (a, b) means a < x < b (open interval, excludes endpoints), [a, b) means a ≤ x < b (half-open). Use ∞ for unbounded intervals: (-∞, ∞) means all real numbers. Use ∪ for union of intervals.

Question 4

How to find domain of rational functions?

Accepted Answer

For rational functions f(x) = p(x)/q(x), the domain excludes values where the denominator equals zero. Set q(x) = 0 and solve for x. These x-values are excluded from the domain (vertical asymptotes). The domain is all real numbers except these values, written in interval notation with union symbols.

Question 5

How to find domain of radical functions?

Accepted Answer

For square root functions √(expression), the radicand (expression under the radical) must be non-negative. Set expression ≥ 0 and solve for x. This gives the domain. For even roots (√, ∜), require non-negative radicand. For odd roots (∛, ∛), all real numbers are allowed.

Question 6

What is the domain and range of logarithmic functions?

Accepted Answer

For logarithmic functions f(x) = log(x) or ln(x), the domain is x > 0 (positive real numbers only), written as (0, +∞). The range is all real numbers (-∞, +∞) because logarithms can output any real value. For log(g(x)), set g(x) > 0 to find the domain.

Question 7

How to find range of polynomial functions?

Accepted Answer

For polynomials: linear functions have range (-∞, +∞). Quadratic and even-degree polynomials have restricted range [yₘᵢₙ, +∞) or (-∞, yₘₐₓ] depending on leading coefficient. Odd-degree polynomials have range (-∞, +∞). Find minimum/maximum using calculus or completing the square for quadratics.

Question 8

What is the domain and range of exponential functions?

Accepted Answer

For exponential functions f(x) = aˣ where a > 0, the domain is all real numbers (-∞, +∞). The range is positive real numbers (0, +∞) because exponential functions never reach zero or become negative. Transformations may shift these intervals.

Question 9

How do you find domain and range from a graph?

Accepted Answer

From a graph: Domain is the horizontal extent (all x-values covered), found by looking left-to-right. Range is the vertical extent (all y-values covered), found by looking bottom-to-top. Identify holes, asymptotes, or endpoints. Open circles indicate excluded values, closed circles indicate included values.

Question 10

What are common domain restrictions?

Accepted Answer

Common restrictions: 1) Division by zero - denominator cannot equal zero. 2) Square roots - radicand must be ≥ 0 for real numbers. 3) Logarithms - argument must be > 0. 4) Inverse trig functions - have specific domains like [-1, 1] for arcsin and arccos. Always check these when finding domain.

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