280 Two-by-Two Linear Graphs
becomes the independent variable and x becomes the dependent variable. We call this
inverse function f−^1 , and state it as follows:
f^ −^1 ( y)=ny+c
where n is the slope and c is the x-intercept. Now if f is a function, f^ −^1 is almost always
a function. But sometimes it isn’t. (Some texts will say, under such conditions, that f
has no inverse, or that f^ −^1 does not exist. Others, such as this book, will say that if we
considerf as a relation, then f always has an inverse relation, but that relation might
not be a function.) Under what conditions is the inverse relation of a linear function
not another function? Here’s a hint: A straight line in Cartesian coordinates represents a
function if and only if its slope is defined.- Draw graphs of three linear functions in Cartesian coordinates whose inverses are
relations, but not functions. - Derive a version of f^ −^1 as we defined it in Prob. 8, but that contains the constants m
andb instead of the constants n and c. Here’s a hint: state f as
y=mx+b
and morph this into SI form with x alone on the left side of the equals sign. Then, in
place of the isolated x, write f^ −^1 (y).