Functions Defined by Equations
Functions defined by equations is a Grade 7 math skill from Yoshiwara Intermediate Algebra where algebraic equations in x and y define a function by expressing y explicitly in terms of x (or implicitly). Students determine if an equation represents a function and use it to evaluate specific outputs.
Key Concepts
Property An equation can define a function by providing a formula to calculate the output value for any given input value. For example, in the equation $h = 1776 16t^2$, for any value of the input variable $t$, a unique value of the output variable $h$ can be determined. We say that $h$ is a function of $t$.
Examples The equation $P = 2l + 2w$ defines the perimeter of a rectangle as a function of its length and width. However, if width is fixed at 5, $P(l) = 2l + 10$ defines perimeter as a function of length.
The equation $y = 4x + 3$ defines a linear function. For any $x$ value you choose, you can find a unique corresponding $y$ value.
Common Questions
How does an equation define a function?
If you can solve the equation for y as a unique expression in x, it defines a function. For example, y = x^2 + 1 defines a function because each x gives exactly one y.
Does x^2 + y^2 = 25 define a function?
No. For most x-values, there are two y-values (positive and negative), so the circle does not define a single-valued function.
How do you evaluate a function given by an equation?
Substitute the given x-value into the equation and solve for y.
What is an implicit function?
An implicit function is defined by an equation without y isolated, like x^2 + y^2 = r^2. You can sometimes extract explicit functions by solving for y.