Domain and Range of a Graph Calculator

Domain and Range Calculator

Select a function type and optionally adjust its shifts to determine its domain and range.

Function Type: Polynomial Function (e.g., x², x³) Rational Function (e.g., 1/x) Square Root Function (e.g., √x) Logarithmic Function (e.g., log(x)) Exponential Function (e.g., 2^x) Absolute Value Function (e.g., |x|)

Horizontal Shift (h): (e.g., for f(x-h))

Vertical Shift (k): (e.g., for f(x)+k)

Results:

Example Function:

Domain:

Range:

Understanding Domain and Range of a Graph

The domain and range are fundamental concepts in mathematics, especially when working with functions and their graphical representations. They describe the set of all possible input values (domain) and output values (range) for a function.

What is the Domain?

The domain of a function refers to all the possible values that the independent variable (usually 'x') can take. In simpler terms, it's the set of all valid inputs for which the function produces a real number output. When looking at a graph, the domain represents the extent of the graph along the horizontal (x-axis).

Common restrictions on the domain arise from:

Division by zero: The denominator of a fraction cannot be zero.
Even roots of negative numbers: You cannot take the square root (or any even root) of a negative number in the real number system.
Logarithms of non-positive numbers: The argument of a logarithm must be strictly positive.

What is the Range?

The range of a function refers to all the possible values that the dependent variable (usually 'y' or f(x)) can produce as output. When looking at a graph, the range represents the extent of the graph along the vertical (y-axis).

Determining the range can sometimes be more challenging than finding the domain, as it often requires understanding the function's behavior, including its maximum or minimum values, asymptotes, and end behavior.

How to Determine Domain and Range for Different Function Types

1. Polynomial Functions (e.g., \(f(x) = x^2 – 3x + 2\))

Domain: For all polynomial functions, the domain is always all real numbers, or \((-\infty, \infty)\). There are no restrictions like division by zero or square roots of negative numbers.
Range:
- If the degree of the polynomial is odd (e.g., \(x^3\)), the range is also all real numbers, \((-\infty, \infty)\).
- If the degree is even (e.g., \(x^2\)), the range will be restricted. It will either be \([k, \infty)\) (if the graph opens upwards) or \((-\infty, k]\) (if the graph opens downwards), where \(k\) is the y-coordinate of the vertex (minimum or maximum point).

Example: For \(f(x) = (x-1)^2 + 2\), the domain is \((-\infty, \infty)\) and the range is \([2, \infty)\).

2. Rational Functions (e.g., \(f(x) = \frac{1}{x-3}\))

Domain: The domain includes all real numbers except for values of \(x\) that make the denominator zero. To find these, set the denominator equal to zero and solve for \(x\).
Range: The range includes all real numbers except for values corresponding to horizontal asymptotes or holes in the graph.

Example: For \(f(x) = \frac{1}{x-2} + 1\), the denominator is zero when \(x=2\), so the domain is \((-\infty, 2) \cup (2, \infty)\). The horizontal asymptote is \(y=1\), so the range is \((-\infty, 1) \cup (1, \infty)\).

3. Square Root Functions (e.g., \(f(x) = \sqrt{x+4}\))

Domain: The expression under the square root (the radicand) must be greater than or equal to zero. Set the radicand \(\ge 0\) and solve for \(x\).
Range: For a standard square root function \(f(x) = \sqrt{g(x)}\), the range is typically \([0, \infty)\). If there's a vertical shift \(+k\), the range becomes \([k, \infty)\). If there's a negative sign in front (e.g., \(-\sqrt{g(x)}\)), the range would be \((-\infty, 0]\) or \((-\infty, k]\).

Example: For \(f(x) = \sqrt{x-3} + 5\), the domain is \(x-3 \ge 0 \Rightarrow x \ge 3\), so \([3, \infty)\). The range is \(y \ge 5\), so \([5, \infty)\).

4. Logarithmic Functions (e.g., \(f(x) = \log(x-1)\))

Domain: The argument of the logarithm must be strictly greater than zero. Set the argument \(> 0\) and solve for \(x\).
Range: For all logarithmic functions, the range is always all real numbers, or \((-\infty, \infty)\).

Example: For \(f(x) = \ln(x+2) – 4\), the domain is \(x+2 > 0 \Rightarrow x > -2\), so \((-2, \infty)\). The range is \((-\infty, \infty)\).

5. Exponential Functions (e.g., \(f(x) = 2^x\))

Domain: For all exponential functions, the domain is always all real numbers, or \((-\infty, \infty)\).
Range: For a standard exponential function \(f(x) = b^x\) (where \(b > 0, b \ne 1\)), the range is \((0, \infty)\). If there's a vertical shift \(+k\), the range becomes \((k, \infty)\).

Example: For \(f(x) = 3^{x-1} + 2\), the domain is \((-\infty, \infty)\). The horizontal asymptote is \(y=2\), so the range is \((2, \infty)\).

6. Absolute Value Functions (e.g., \(f(x) = |x|\))

Domain: For all absolute value functions, the domain is always all real numbers, or \((-\infty, \infty)\).
Range: For a standard absolute value function \(f(x) = |g(x)|\), the range is typically \([0, \infty)\). If there's a vertical shift \(+k\), the range becomes \([k, \infty)\). If there's a negative sign in front (e.g., \(-|g(x)|\)), the range would be \((-\infty, 0]\) or \((-\infty, k]\).

Example: For \(f(x) = |x+5| – 3\), the domain is \((-\infty, \infty)\). The vertex is at \((-5, -3)\) and the graph opens upwards, so the range is \([-3, \infty)\).

Using the calculator above, you can select different function types and observe how horizontal (h) and vertical (k) shifts impact their domain and range, helping you visualize these fundamental concepts.