Calculus Examples

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

Step 1

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

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

Step 2

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

$x \approx 1.73920386$

Step 3

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

Interval Notation:

$(- \infty, 1.73920386) \cup (1.73920386, \infty)$

Set-Builder Notation:

${x | x \neq 1.73920386}$

Step 4

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

Interval Notation:

$[- 0.53600628, \infty)$

Set-Builder Notation:

${y | y \geq - 0.53600628}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 1.73920386) \cup (1.73920386, \infty), {x | x \neq 1.73920386}$

Range: $[- 0.53600628, \infty), {y | y \geq - 0.53600628}$

Step 6