Is it a Function or not??

Question

My teacher (for Algebra 2) taught us how to do functions without calculators. We didn't get into the graphing part, but just the basics.

{ (x, y) | y=x^2+3 }

Domaine: { x| x is an element of real numbers}

Range: { y| y is_> ((less than/ equal to)) 3 }

Function: yes

My problem is: how do you know that y is less than/ equal to 3? In all of the problems we did, it came out that y was less than/ equal to whatever number was in the problem ( i.e. 3 was in the problem with x).

And how do you tell if it's a function or not?? I know the rule "a set of ordered pairs such that for all members of the domain there exists exactly one member of the range."

Answer 1

{ (x, y) | y=x^2+3 }.

Positive numbers squared give positive numbers.

Negative numbers squared give positive numbers.

Zero squared gives zero.

Therefore:

x^2 >= 0, whatever the value of x,

and

x^2 + 3 >= 3 (that is greater than or equal to 3, not less than or equal).

A formula represents a function if no vertical line cuts its graph more than once. The graph of y = x^2 + 3 is a parabola opening upwards, and the formula therefore represents a function.

If you consider

y^2 = x - 3,

that is a horizontal parabola, and a vertical line can cut it more than once.

y is not a function of x, because:

y = +/- sqrt(x - 3) clearly has two values for y when x > 3.

If you change the formula to:

y = sqrt(x - 3),

allowing only the positive square root, then y is a function of x.