the range is the entire set of real numbers. That means there is more than one value in the
range for the single value in the domain, causing the inverse to fail the “function test.”
- Figure B-9 shows graphs of three linear functions whose inverses are relations but not
functions. We’ve given the original functions the arbitrary names f(x),g(x), and h(x).
(There are infinitely many other examples, of course.) The equations always take the form
y= 0 x+b
where b is the y-intercept. The slope is always 0, and the y-intercept can be any real num-
ber. In these examples, x is the independent variable and y is the dependent variable. Axis
increments are not indicated, because it doesn’t matter what they are! When the variables
are transposed to obtain the inverse relations, the lines become parallel to the dependent-
variable axis. The inverse relations always fail the “function test,” because there is more
than one value in the range for the single value in the domain.
- Let’s start with the general SI form of a linear equation, as the hint suggests. The
function can then be stated as
y=mx+b
with the understanding that y=f(x). We want to morph this into SI form, treating y as
the independent variable and x as the dependent variable. Let’s begin by subtracting b from
each side. That gives us
y−b=mx
Chapter 17 649
x
y
f(x)
g(x)
h(x)
Figure B-9 Illustration for the solution to Prob. 9 in Chap. 17.
