Answer: Option "a" is the only expression that represents a function.
Explanation:
A function f(x) = y is a "operator" that takes an input element, x, and assigns it to only one output element, y.
So, if we have that for a given value of x.
f(x) = y and f(x) = h
where y and h are different values, then this is not a function, because is assigning the input value x to two different output values.
Let's see the different options:
a) {(1, 2), (4, −2), (8, 3), (9, −3)}
This points are of the form (x, y)
We can see that each value of x is assigned to only one value of y, so this can represent a function.
b) y^2 = 16 − x^2
Ok, suppose that x = 0, then:
y^2 = 16 - 0 = 16
then we have that y*y = 16.
So y can take two different values:
y = 4 ---> 4*4 = 16
y = -4 ---> -4*-4 = 16.
So this is not a function.
c) 2x^2 + y^2 = 5
First, we want to isolate y in one side:
y^2 = 5 - 2*x^2
Here we have a similar case to the option b, and we can use a similar argument to prove that this is not a function, so we can discard this.
d) x = 7.
Ok, this is not a relation between two variables, so this is not a function, as if x is the input value, we have only one value of x that solves the equation.