Explain how you can tell whether a polynomial equation is a function and not just a relation.
easy. All polynomials are functions.
each value of x produces a single value of y.
This assuming I am correctly interpreting what you mean by a "polynomial equation."
Is this true:
for each value of the independent variable, is there one and only one value for the dependent variable?
For each value of the independent there is either none or one value for the dependent right? Meaning that x cannot have 2 y values and then its a function. (Passes VLT)
if there is no value of y for some x, then it is not even a relation, much less a function.
There must be exactly one y for each x.
To determine whether a polynomial equation is a function and not just a relation, you need to understand the fundamental difference between the two concepts.
A relation is a set of ordered pairs where each input value (x) is associated with one or more output values (y). It means that for a given x-value, there can be multiple y-values associated with it. In other words, a relation does not have the restriction that every x-value must have a unique y-value.
On the other hand, a function is a special type of relation where each input value (x) is associated with exactly one unique output value (y). This means that for every x-value, there can only be a single y-value associated with it.
Now, let's apply this understanding to polynomial equations. A polynomial equation is an equation that involves variables raised to non-negative integer powers and combined using addition, subtraction, and multiplication operations. It has the general form:
f(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_1 * x + a_0
Here, the coefficients (a_n, a_(n-1), ..., a_0) are constants, and the powers of x (n, n-1, ...) define the degree of the polynomial.
To determine whether a polynomial equation represents a function or just a relation, you need to check if it satisfies the definition of a function by ensuring that each input value (x) has a unique output value (y).
One way to assess this is by examining the degree of the polynomial equation. Polynomials of degree 1 have a linear relationship, where each x-value corresponds to a single y-value. Polynomials of higher degrees, such as quadratic, cubic, or higher, also represent functions because each x-value is associated with only one y-value.
However, if you encounter an equation where the degree of the polynomial is not defined, or if there is any term involving a fractional power or radical (such as square root), it may indicate a non-function relation. In such cases, you may need to perform additional analysis or mathematical techniques, like graphing or domain/range analysis, to determine whether the equation represents a function.
In summary, to determine if a polynomial equation is a function or just a relation, you need to ensure that each x-value is associated with a unique y-value. If each input value has only one associated output value, the polynomial equation is a function.