AP Calculus AB: 8.4.3 Domain-Restricted Functions and the Derivative

If the variable of a function is raised to a fractional exponent that has an even denominator, the function may not be defined for all real numbers.
• To graph a function:
1. Find critical points using the first derivative.
2. Determine where the function is increasing or decreasing.
3. Find inflection points using the second derivative.
4. Determine where the function is concave up or concave down.
• On an interval, the sign of the first derivative indicates whether the function is increasing or decreasing. The sign of the second derivative indicates whether the function is concave up or concave down

• If a function involves a fractional exponent that has an even denominator even when reduced, then the domain of the function may be restricted. This particular function is not defined for x-values greater than 6. Keep this in mind as you go through the steps for sketching its graph.

• First, take the derivative using the product rule. Set it equal to zero to find the critical points. Then make a sign chart for first derivative. It is not defined for x-values greater than or equal to 6.

• Next, take the second derivative, which requires the quotient rule. Set it equal to zero and solve for x. The only solution is x = 8, but that is not in the domain, so there are no inflection point candidates.

• Now make a sign chart for the second derivative.

• Once you have analyzed the function, sketch its graph.

• The function is increasing to the left of x = 4 and decreasing from x = 4 to x = 6. It has a maximum at x = 4. There is an endpoint minimum at x = 6 because the function is decreasing at the end of its domain.

• The graph is concave down everywhere, so there is no
  inflection point.

The graph of y=x−x−1+2y = x - \sqrt{x-1} + 2y=x−x−1​+2 is defined for x≥1x \geq 1x≥1 and looks like a curve starting at (1,2)(1, 2)(1,2) increasing gradually, resembling a line shifted and slightly curved downward near x=1x=1x=1.

The best sketch of f(x)=8−xf(x) = \sqrt{8 - x}f(x)=8−x​ is a decreasing curve starting at (0,8)(0, \sqrt{8})(0,8​) and ending at (8,0)(8, 0)(8,0), defined for x≤8x \leq 8x≤8.

Domains must be restricted so that any cusp points are excluded.

The best sketch of f(x)=x2−1f(x) = \sqrt{x^2 - 1}f(x)=x2−1​ is two branches starting from x=±1x = \pm 1x=±1, increasing symmetrically outward, with a gap between −1-1−1 and 111 where the function is undefined.