How do you use the vertical line test to show #x^2=xy-1# is a function?

Answer 1

#f(x)=x+1/x#

#x^2=xy-1#
Let's solve in terms of #y# so it is in standard function form:
#x^2+1=xy#
#x+1/x=y#
#y=x+1/x#

This seems like a function, so let's examine the graph:

graph{y=x+1/x [-5, 5, 10, 10]}

if you draw a vertical line anywhere across the domain of this graph (all #x# values) does it cross the graph twice? No, so this meets the criteria and is a function.
#f(x)=x+1/x#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

Apply the vertical line test by drawing vertical lines through the graph. If each vertical line intersects the graph at most once, then x^2 = xy - 1 is a function.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To use the vertical line test to show that (x^2 = xy - 1) is a function, we need to examine whether any vertical line intersects the graph of the equation (x^2 = xy - 1) at more than one point. If every vertical line intersects the graph at most once, then the equation represents a function.

Rearranging the given equation, we get: [ x^2 - xy + 1 = 0 ]

This equation represents a quadratic function in terms of (x). The vertical line test states that if a vertical line intersects the graph of a function at more than one point, then the relation is not a function.

By applying the vertical line test to the graph of the equation (x^2 - xy + 1 = 0), we can observe that every vertical line intersects the graph at most once. Therefore, the equation (x^2 = xy - 1) represents a function.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7