### Video Transcript

Which of the following functions
does not have an inverse over its whole domain? Assume that the codomain of each
function is equal to its range. Is it option (A) π of π₯ is equal
to two π₯? Option (B) π of π₯ is equal to two
to the power of π₯. Option (C) π of π₯ is equal to one
over π₯. Or option (D) π of π₯ is equal to
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 our four
functions isnβt one-to-one by sketching graphs of each of them.

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.

Letβs now consider option (B). 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. We have therefore ruled out options
(A) and (B). Now, letβs consider option (C).

For the graph of π¦ equals one over
π₯, there are 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. The π₯- and π¦-axes are horizontal
and vertical asymptotes. And we can once again see that by
drawing horizontal and vertical lines, the function is one to one over its entire
domain.

This leaves us with just option
(D). 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.

The correct answer is therefore
option (D). The function that does not have an
inverse over its whole domain is π of π₯ is equal to π₯ squared.