The domain of a function refers to the set of all possible input values \( x \) that will produce a real output in the function. To find the domain of a function defined by an equation, we need to check for any restrictions like division by zero, square roots of negative numbers, or logarithms of non-positive numbers.
When a function is defined by an equation, the domain includes all 𝑥-values for which the equation produces a real number as output.
Consider the linear function:
$$ f(x) = 3x + 5 $$
There are no restrictions, so the domain is:
$$ \text{Domain: } (-\infty, \infty) $$
For the function:
$$ f(x) = \frac{2x + 3}{x - 1} $$
The denominator must not be zero, so \( x = 1 \) is excluded. The domain is:
$$ \text{Domain: } (-\infty, 1) \cup (1, \infty) $$
For the square root function:
$$ f(x) = \sqrt{x - 4} $$
The expression under the square root must be non-negative, so the domain is:
$$ \text{Domain: } [4, \infty) $$
For the logarithmic function:
$$ f(x) = \log(x - 2) $$
The argument of the logarithm must be positive, so the domain is:
$$ \text{Domain: } (2, \infty) $$
For the function:
$$ f(x) = \frac{\sqrt{x - 1}}{x + 2} $$
We need to check both the square root and the denominator:
Thus, the domain excludes 𝑥 = − 2 and requires 𝑥 ≥ 1. Therefore, the domain is:
$$ \text{Domain: } [1, \infty) $$