Precalculus Examples

$f (x) = \frac{2 x^{3} - 3 x + 1}{x^{3} - 5 x + 7}$

Step 1

Set the denominator in $\frac{2 x^{3} - 3 x + 1}{x^{3} - 5 x + 7}$ equal to $0$ to find where the expression is undefined.

$x^{3} - 5 x + 7 = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 2.74734654$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 2.74734654) \cup (- 2.74734654, \infty)$

Set-Builder Notation:

${x | x \neq - 2.74734654}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 2.74734654) \cup (- 2.74734654, \infty), {x | x \neq - 2.74734654}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6