Vertical Line Test

[nycmath]

Precalculus
Michael Sullivan
Edition 10
Chapter 2, Section 2.2

A. Select two favorite functions to determine if the graph is the graph of a function by using the vertical line test.

B. If it is a function, use the graph to find:

1. The domain and range
2. The intercepts, if any
3. Any symmetry with respect to the x-axis, y-axis, or the origin.

See attachment.

[Math Tutor]

I'll take graphs #12 and #19.

A. Vertical line test
A graph is the graph of a function exactly when no vertical line crosses it more than once (one input, one output). Sliding a vertical line across #12 and across #19, it always meets the curve in exactly one point, so both are functions. (For contrast, #11, #15 and #16 all fail the test - a vertical line hits them twice.)

B1. Graph #12 (the curve rising left to right, flattening onto the x-axis on the left)

1. Domain and range. The arrows say the curve continues forever both ways, so the domain is [tex]\{x\,|\,x \in \mathbb{R}\}=(-\infty,\infty)[/tex]. Going left the curve gets closer and closer to the x-axis but never reaches it, so [tex]y=0[/tex] is a horizontal asymptote and the range is [tex]\{y\,|\,y>0\}=(0,\infty)[/tex].

2. Intercepts. It crosses the y-axis at [tex](0,1)[/tex]. There is no x-intercept, since the graph never actually touches [tex]y=0[/tex].

3. Symmetry. None. The left half is nothing like the right half, so there is no y-axis symmetry and no origin symmetry.

B2. Graph #19 (the "plateau": rises to [tex](-1,2)[/tex], flat across to [tex](1,2)[/tex], then falls)

1. Domain and range. Again the arrows continue forever, so the domain is [tex](-\infty,\infty)[/tex]. The highest point reached is [tex]y=2[/tex] (the whole flat top), and the graph drops without bound on both ends, so the range is [tex]\{y\,|\,y \le 2\}=(-\infty,2][/tex].

2. Intercepts. The graph meets the x-axis at [tex](-3,0)[/tex] and [tex](3,0)[/tex], and it meets the y-axis at [tex](0,2)[/tex].

3. Symmetry. Symmetric with respect to the y-axis: whatever happens at [tex]x[/tex] happens at [tex]-x[/tex] (for example [tex](-1,2)[/tex] and [tex](1,2)[/tex], [tex](-3,0)[/tex] and [tex](3,0)[/tex]), so [tex]f(-x)=f(x)[/tex] and the function is even. Not symmetric about the origin, and not about the x-axis.

One useful remark for part 3 in general: apart from the trivial case, no graph of a function can be symmetric with respect to the x-axis, because reflecting a point [tex](a,b)[/tex] to [tex](a,-b)[/tex] would put two points on the same vertical line - exactly what the vertical line test forbids.

[nycmath]

I'll take graphs #12 and #19.

A. Vertical line test
A graph is the graph of a function exactly when no vertical line crosses it more than once (one input, one output). Sliding a vertical line across #12 and across #19, it always meets the curve in exactly one point, so both are functions. (For contrast, #11, #15 and #16 all fail the test - a vertical line hits them twice.)

B1. Graph #12 (the curve rising left to right, flattening onto the x-axis on the left)

1. Domain and range. The arrows say the curve continues forever both ways, so the domain is [tex]\{x\,|\,x \in \mathbb{R}\}=(-\infty,\infty)[/tex]. Going left the curve gets closer and closer to the x-axis but never reaches it, so [tex]y=0[/tex] is a horizontal asymptote and the range is [tex]\{y\,|\,y>0\}=(0,\infty)[/tex].

2. Intercepts. It crosses the y-axis at [tex](0,1)[/tex]. There is no x-intercept, since the graph never actually touches [tex]y=0[/tex].

3. Symmetry. None. The left half is nothing like the right half, so there is no y-axis symmetry and no origin symmetry.

B2. Graph #19 (the "plateau": rises to [tex](-1,2)[/tex], flat across to [tex](1,2)[/tex], then falls)

1. Domain and range. Again the arrows continue forever, so the domain is [tex](-\infty,\infty)[/tex]. The highest point reached is [tex]y=2[/tex] (the whole flat top), and the graph drops without bound on both ends, so the range is [tex]\{y\,|\,y \le 2\}=(-\infty,2][/tex].

2. Intercepts. The graph meets the x-axis at [tex](-3,0)[/tex] and [tex](3,0)[/tex], and it meets the y-axis at [tex](0,2)[/tex].

3. Symmetry. Symmetric with respect to the y-axis: whatever happens at [tex]x[/tex] happens at [tex]-x[/tex] (for example [tex](-1,2)[/tex] and [tex](1,2)[/tex], [tex](-3,0)[/tex] and [tex](3,0)[/tex]), so [tex]f(-x)=f(x)[/tex] and the function is even. Not symmetric about the origin, and not about the x-axis.

One useful remark for part 3 in general: apart from the trivial case, no graph of a function can be symmetric with respect to the x-axis, because reflecting a point [tex](a,b)[/tex] to [tex](a,-b)[/tex] would put two points on the same vertical line - exactly what the vertical line test forbids.

Wow! Another detailed reply for my notes. Thank you.

[Eigenvalue]

11. Domain: (-infinity, -2) U (-2, infinity)
Range: (-infinity, 0) U (0, infinity)
x-intercepts: (2,0) (-2,0)
y-intercepts: N/A
Symmetrical in respect to the y-axis (even)
equation: x²-y²=1

[Eigenvalue]

12. Domain: (-infinity, infinity)
Range: (0,infinity)
y-intercept (0,1)
no symmetry

[Eigenvalue]

15. not a function
16. not a function

[Eigenvalue]

19. domain:[-3,3]
Range: [0,3]
x-intercepts -(3,0) (3,0)
y-intercept (0,2)
Symmetry in respect to the y-axis

20. Domain: [0,3]
Range: [0,2]
x-intercepts (-3,0) (2,0)
y-intercepts: (0,2)
No symmetry

[nycmath]

11. Domain: (-infinity, -2) U (-2, infinity)
Range: (-infinity, 0) U (0, infinity)
x-intercepts: (2,0) (-2,0)
y-intercepts: N/A
Symmetrical in respect to the y-axis (even)
equation: x²-y²=1

Thank you but you didn't have to answer each one.

[nycmath]

11. Domain: (-infinity, -2) U (-2, infinity)
Range: (-infinity, 0) U (0, infinity)
x-intercepts: (2,0) (-2,0)
y-intercepts: N/A
Symmetrical in respect to the y-axis (even)
equation: x²-y²=1

Thank you but selecting two problems by explaining your steps along the way is better for study notes. Nonetheless, you are wonderful.

[nycmath]

12. Domain: (-infinity, infinity)
Range: (0,infinity)
y-intercept (0,1)
no symmetry

May I make a suggestion? Try to reply to more than one question per reply. This will keep answers concisely written.

[nycmath]

15. not a function
16. not a function

Good to know but why are 15 and 16 not functions? They fail the vertical line test. Yes? No?

[nycmath]

19. domain:[-3,3]
Range: [0,3]
x-intercepts -(3,0) (3,0)
y-intercept (0,2)
Symmetry in respect to the y-axis

20. Domain: [0,3]
Range: [0,2]
x-intercepts (-3,0) (2,0)
y-intercepts: (0,2)
No symmetry

Thank you for your time, effort and contribution.

[Eigenvalue]

15. not a function
16. not a function

Good to know but why are 15 and 16 not functions? They fail the vertical line test. Yes? No?

15 and 16 are not a functions as they fails the vertical line test
One input/x- value must only have 1 output/y-value

[nycmath]

15. not a function
16. not a function

Good to know but why are 15 and 16 not functions? They fail the vertical line test. Yes? No?

15 and 16 are not a functions as they fails the vertical line test
One input/x- value must only have 1 output/y-value

That works for me. Thank you.