Algebra

PROBLEM 3: Determine whether each equation is a function of x or not. a) y = (x^2 - 4) / (x + 2) b) y = sqrt(2x + 5) c) x = y^2 - 4y + 1 d) x^2 + 4y^2 - 6x + 8y - 11 = 0 e) y^3 - 2y = x^2 + 1

Step-by-step solution with explanation

Final Answer

a) y = (x²-4)/(x+2) — IS a function of x (one output per input, x ≠ -2). b) y = √(2x+5) — IS a function of x (principal root gives one output). c) x = y²-4y+1 — NOT a function of x (one x can yield two y values). d) x²+4y²-6x+8y-11=0 — NOT a function of x (ellipse fails vertical line test). e) y³-2y = x²+1 — NOT a function of x (cubic in y can have up to 3 real solutions for one x).

Step-by-step solution

1

Test part a: rational expression

For every x value (except x = -2, where the denominator is zero), exactly one y value is produced. Each input gives at most one output, so this IS a function of x.
2

Test part b: square root expression

The square root symbol means we only take the principal (positive) root. So each valid x gives exactly one y value. This IS a function of x (defined when ).
3

Test part c: x equals quadratic in y

Solving for y gives two possible y values for most x inputs. For example, x = 1 gives , so y = 0 or y = 4. One x maps to two y values, so this is NOT a function of x.
4

Test part d: ellipse equation

Rewrite by completing the square: . This is an ellipse. A vertical line hits an ellipse in two points for most x values, so this is NOT a function of x.
5

Test part e: cubic in y

For a single x, the right side is one number, say . The equation is a cubic, which can have up to 3 real solutions for y. For example, x = 0 gives , which has three real roots. One x can map to multiple y values, so this is NOT a function of x.

Understanding this problem

Learning Insight

A relation is a function of x when every x input produces EXACTLY one y output. The vertical line test captures this visually — if any vertical line crosses the graph more than once, it fails. Algebraically, if solving for y can give more than one answer, it's not a function.

Quick Tip

Quick check: if you can isolate y and get a single expression with no ± sign, it's almost always a function. If you see ± or a quadratic/higher-degree equation in y, suspect it is NOT a function.

Common Mistake

Students often think part (a) is not a function because of the excluded point at x = -2. A single missing point does NOT break the function definition — what matters is whether any allowed x gives two or more y values.