Some functions are defined by mathematical rules or procedures expressed in
equation form. If it is possible to express the function output with a
formula involving the input quantity, then we can define a function in algebraic form. For example, the equation
expresses a functional relationship between
and
We can rewrite it to decide if
is a function of
Given a function in equation form, write its algebraic formula.
Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves
only the input variable.
Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.
Finding an equation of a function
Express the relationship
as a function
if possible.
To express the relationship in this form, we need to be able to write the relationship where
is a function of
which means writing it as
Does the equation
represent a function with
as input and
as output? If so, express the relationship as a function
First we subtract
from both sides.
We now try to solve for
in this equation.
We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function
If we graph both functions on a graphing calculator, we will get the upper and lower semicircles.
Are there relationships expressed by an equation that do represent a function but that still cannot be represented by an algebraic formula?
Yes, this can happen. For example, given the equation
if we want to express
as a function of
there is no simple algebraic formula involving only
that equals
However, each
does determine a unique value for
and there are mathematical procedures by which
can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for
as a function of
even though the formula cannot be written explicitly.
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
