A Function of x

[nycmath]

Precalculus
Michael Sullivan
Edition 10
Chapter 2, Section 2.1

Determine whether the equation defines y as a function of x.

See attachment.

[Math Tutor]

36. [tex]y = \pm\sqrt{1 - 2x}[/tex]

The [tex]\pm[/tex] already tells you there are two outputs for most inputs.

Take [tex]x = 0[/tex]: [tex]y = \pm\sqrt{1} = 1[/tex] or [tex]y = -1[/tex]. Two different y-values for the same x, so a vertical line hits the graph twice and the vertical line test fails.

So the equation does NOT define y as a function of x.

(The only x where you get a single y is [tex]x = \tfrac{1}{2}[/tex], where [tex]y = 0[/tex]; and for [tex]x > \tfrac{1}{2}[/tex] there is no real y at all. One good x with two y's is enough to say "not a function".)

[Eigenvalue]

34. y is a function of x, for every 1 x value there is only 1 y value

36. y is not a function of x, for each x value, there are 2 y values. For example, when x=-4, y=[tex]\pm[/tex]3

[nycmath]

36. [tex]y = \pm\sqrt{1 - 2x}[/tex]

The [tex]\pm[/tex] already tells you there are two outputs for most inputs.

Take [tex]x = 0[/tex]: [tex]y = \pm\sqrt{1} = 1[/tex] or [tex]y = -1[/tex]. Two different y-values for the same x, so a vertical line hits the graph twice and the vertical line test fails.

So the equation does NOT define y as a function of x.

(The only x where you get a single y is [tex]x = \tfrac{1}{2}[/tex], where [tex]y = 0[/tex]; and for [tex]x > \tfrac{1}{2}[/tex] there is no real y at all. One good x with two y's is enough to say "not a function".)

You have a natural talent for numbers.

[nycmath]

34. y is a function of x, for every 1 x value there is only 1 y value

36. y is not a function of x, for each x value, there are 2 y values. For example, when x=-4, y=[tex]\pm[/tex]3

Perfect. Easy. To the point without delay.