### Video Transcript

Which of the following functions
does not have an inverse over its whole domain? (a) π of π₯ equals two π₯. (b) π of π₯ equals two to the
power of π₯. (c) π of π₯ equals one over
π₯. Or (d) π of π₯ equals π₯
squared.

To answer this question, we need to
consider the conditions under which a function has an inverse. A function only has an inverse over
its entire domain if it is a one-to-one function. This means that every input to the
function produces a unique output so that when we go the other way using the inverse
function, every input to this function also produces a unique output. We can work out which of these
functions isnβt one-to-one by sketching graphs of each function. For option (a), the graph of π¦
equals π of π₯ or π¦ equals two π₯ is a straight line graph through the origin. This is clearly a one-to-one
function over its entire domain. Every input has a unique
output. And so going the other way, the
same will be true.

More generally though, we can
determine whether a function is one-to-one by drawing horizontal and vertical lines
on its graph. If each horizontal and each
vertical line intersects the graph a maximum of one time, then the function is
one-to-one. The graph of π¦ equals two to the
power of π₯ is an exponential graph passing through the point zero, one with a
horizontal asymptote along the π₯-axis. This is also a one-to-one function
over its entire domain because every vertical line intersects the graph once and
every horizontal line intersects the graph a maximum of one time.

For the graph of π¦ equals one over
π₯, thereβre two branches, one in the first quadrant, where π₯ and π¦ are both
positive, and one in the third quadrant, where π₯ and π¦ are both negative. We can see, though, that the
function is one-to-one over its entire domain. That just leaves option (d)
then. The graph of π¦ equals π₯ squared
is a positive parabola with its minimum point at the origin. We can see that if we draw
horizontal lines across the graph, some of these lines will intersect the graph in
more than one place. This means that for certain output
values, there are two possible input values, and so the function π of π₯ isnβt
one-to-one. This means we canβt find the
inverse of π of π₯ over its entire domain because we donβt know which of those
input values the output value should be mapped to by the inverse function. Our answer is that the function
which does not have an inverse over its whole domain is π of π₯ equals π₯
squared.