Examples with Detailed Solutions
Example 1: Domain of a Rational Function
Find the domain of function \( f \) defined by:
\[ f(x) = \dfrac{1}{x-1} \]Solution:
\( x \) can take any real number except 1 since \( x = 1 \) would make the denominator equal to zero, and division by zero is not allowed in mathematics. Hence, the domain in interval notation is given by:
\[ (-\infty, 1) \cup (1, +\infty) \]Example 2: Domain of a Radical Function
Find the domain of function \( f \) defined by:
\[ f(x) = \sqrt{2x-8} \]Solution:
The expression defining function \( f \) contains a square root. The expression under the radical must satisfy the condition for the function to take real values:
\[ 2x - 8 \geq 0 \]Solving this linear inequality:
\[ 2x \geq 8 \implies x \geq 4 \]The domain, in interval notation, is:
\[ [4, +\infty) \]Example 3: Domain of a Rational and Radical Function
Find the domain of function \( f \) defined by:
\[ f(x) = \dfrac{\sqrt{-x}}{(x-3)(x+5)} \]Solution:
1. The expression under the square root must satisfy:
\[ -x \geq 0 \implies x \leq 0 \]2. The denominator must not be zero:
\[ (x-3)(x+5) \neq 0 \implies x \neq 3 \text{ and } x \neq -5 \]Combining these conditions (\( x \leq 0 \) and excluding \( x = -5 \)), the domain of \( f \) is:
\[ (-\infty, -5) \cup (-5, 0] \]Example 4: Range of a Quadratic Function
Find the range of function \( f \) defined by:
\[ f(x) = x^2 - 2 \]Solution:
The domain of this function is the set of all real numbers. The range is the set of values that \( f(x) \) takes as \( x \) varies. If \( x \) is a real number, \( x^2 \) is either positive or zero:
\[ x^2 \geq 0 \]Subtracting 2 from both sides:
\[ x^2 - 2 \geq -2 \]This indicates that \( x^2 - 2 \) takes all values greater than or equal to -2. The range of \( f \) is:
\[ [-2, +\infty) \]A graph of \( f \) also helps interpret the range. Note that the lowest point on the parabola has a \( y \)-value of -2.
