4.2: Implicitly Defined Curves | AP Precalculus

How it works

An implicit equation relates coordinates without necessarily giving one output for each input. Test whether a vertical line can meet the curve more than once. Choosing a branch can create a function but also restricts its range.

F(x,y)=0,\quad x^2+y^2=9\ \Rightarrow\ y=\pm\sqrt{9-x^2}

Worked examples

Example 1

Does $\displaystyle x^2+y^2=9$ define y as a single function of x?

No. At $\displaystyle x=0$ , both $\displaystyle y=3$ and $\displaystyle y=-3$ satisfy the equation.

Example 2

Write the upper semicircle as a function.

$\displaystyle y=\sqrt{9-x^2}$ with domain $\displaystyle [-3,3]$ and range $\displaystyle [0,3]$ .

Problems

Problem 4.2.1

For $\displaystyle x^2+y^2=9$ , find all y-values when $\displaystyle x=0$ and when $\displaystyle x=3$ .

Circle centered at the origin with radius 3. — The full circle fails the vertical-line test for a function of x.

Show worked solution

At $\displaystyle x=0$ , $\displaystyle y=\pm3$ . At $\displaystyle x=3$ , $\displaystyle y=0$ . The first input fails the uniqueness requirement for a function.

Circle centered at the origin with radius 3. The solution adds labeled relationships. — At x = 0 the two outputs are 3 and -3.

Problem 4.2.2

Solve $\displaystyle xy=6$ for y and state the domain.

Show worked solution

$\displaystyle y=6/x$ , with $\displaystyle x\ne0$ . This relation does define a function of x on that domain.

Problem 4.2.3

Does $\displaystyle x=y^2$ define y as a function of x over all its real points?

Show worked solution

No. For $\displaystyle x>0$ there are two outputs $\displaystyle \pm\sqrt{x}$ . Restricting to $\displaystyle y\ge0$ gives one branch.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.2.1

Problem 4.2.2

Problem 4.2.3

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