Implicitly Defined Functions
Introduced| English |
|---|
| implicitly defined/ɪmˈplɪsɪtli dɪˈfaɪnd/ |
| conic section/ˈkɒnɪk ˈsekʃn/ |
When y refuses to stand alone
- Most functions we meet are explicit: $y$ sits alone, equal to some formula in $x$.
- But an equation like $x^2 + y^2 = 25$ tangles $x$ and $y$ together.
- You cannot cleanly write "$y = \dots$" — yet the equation still describes a perfect curve.
- Such a relationship is defined implicitly, and it opens up a whole family of shapes.
Explicit versus implicit
- An explicit function isolates the output: $y = x^2 - 3$.
- An implicitly defined 隐式定义 relation leaves $x$ and $y$ mixed: $x^2 + y^2 = 25$.
- The implicit form states a condition that points on the curve must satisfy.
- It is often the most natural way to describe a shape.
An implicitly defined relationship is one where…
An implicitly defined curve, like $x^2 + y^2 = 25$, mixes $x$ and $y$ without isolating $y$.
A circle, implicitly
- The equation $x^2 + y^2 = 25$ says "the distance from the origin is $5$".
- Every point obeying it lies on a circle of radius $5$ — a classic conic section 圆锥曲线.
- No rearranging is needed to see the shape; the equation already captures it.
- Ellipses, parabolas, and hyperbolas are all defined this same implicit way.
The equation $x^2 + y^2 = 25$ describes…
All points at distance $5$ from the origin satisfy $x^2 + y^2 = 25$ — a circle of radius $5$.
An implicit relation like $x^2 + y^2 = 25$ need not be a function of $x$.
Most $x$-values give two $y$-values (top and bottom of the circle), so it fails the vertical-line test.
Select all true statements about implicitly defined relations.
Many implicit relations cannot be solved neatly for $y$ at all. The other three are correct.
Not a function of x
- For most $x$-values, an implicit curve gives two $y$-values (top and bottom).
- So it usually fails the vertical-line test — it is a relation, not a function.
- $x^2 + y^2 = 25$ solved for $y$ is $y = \pm\sqrt{25 - x^2}$: two semicircle branches.
- Each branch on its own is a function; together they form the whole curve.
Explicit or implicit?
An implicit equation relates x and y without solving for y. Sort each.
Solving $x^2 + y^2 = 25$ for $y$ gives $y = \pm\sqrt{25 - x^2}$ — that is ____ separate branches.
The $\pm$ splits the circle into an upper and a lower semicircle — two explicit functions.
Working with implicit curves
- To find points, substitute one coordinate and solve for the other.
- To sketch, recognise the shape (circle, ellipse, …) from the equation's form.
- Some implicit relations cannot be solved for $y$ at all — you work with them as they are.
- In calculus, implicit differentiation finds slopes without ever isolating $y$.
An implicit equation is a relation, not automatically a function. Do not assume you can write $y = f(x)$ — for a circle you get $y = \pm\sqrt{25 - x^2}$, and forgetting the $\pm$ silently drops half the curve.
Find the points on $x^2 + y^2 = 25$ where $x = 3$.
- Substitute: $9 + y^2 = 25$, so $y^2 = 16$.
- Then $y = \pm 4$ — two points, $(3, 4)$ and $(3, -4)$.
- One $x$-value, two $y$-values: the mark of a non-function relation.
An implicitly defined relation mixes $x$ and $y$ in one equation, like $x^2 + y^2 = 25$, rather than isolating $y$. It can describe circles and other conic sections, usually is not a function of $x$ (solving gives $\pm$ branches), and is often the most natural way to state a curve.