Recognizing Functions

Question

Identify which of the following are examples of y as a function of x, and give a brief explanation either analytically or graphically.

For those that are not functions, provide a specific value of x where two (or more) values of y can be assigned.

a) 3x²-y=5

b) x²+y²=4

c) y=5e^-3x

d) x=√(y-1)

e) sin y = x

Answer 1

one method is use a graphing calculator. then apply the vertical line test. If a vertical line can cut the graph in more than one place, it's Not a function.

a) is an upward opening parabola, which is a function, but not a one to one function, as it fails the horizontal line test.

b) is a circle, center at the origin, with radius 2, and not a function

for x = 1, there are two y values +/-square root of 3, about +1.732 and -1.732

c) is an exponential function, x axis is its horizontal asymptote. It's a one to one function

d) is the upper half of a horizontal parabola and a one to one function

e) is an inverse sine wave, y=sin^-1(x), and not a function. y=sinx is a function. siny=x is the sinx function reflected over the 45 degree line, having an infinite number of intersections with a vertical line sin90 = sin450 = 1