Question

Given the function f described by f(x)=2x-3, prove that f is one-to-one.

Answer 1

To show that a function is one-to-one (injective)
we need to show:
\[f(x_1) = f(x_2) \Rightarrow x_1 = x_2\]

So we just plug in x1 and x2 and get:
\[f(x_1)= f(x_2) \Rightarrow 2x_1-3=2x_2-3 \Rightarrow 2x_1=2x_2 \Rightarrow x_1 = x_2\]

Answer 2

f(x) is one-to-one if and only if f(a) = f(b) implies a=b (and vice versa, ie a = b implies f(a) = f(b))


So say you have two variables p and q. Now say that they're equal. So p = q


So f(p) = 2p-3 and f(q) = 2q-3, but since p=q, we know that f(q) = 2p-3 as well. So f(p) = f(q) when p = q


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Similarly, if f(x) = f(y), then 2x-3 = 2y-3, which means that x = y (when you simplify)


So these two facts show us that f(x) is one-to-one.

Answer 3

A visual way of seeing this is that the graph of f(x) passes BOTH the vertical line test AND the horizontal line test. If it passes both tests, then the function is one-to-one.

Answer 4

nice subscripts.

Answer 5

thank you ALL!!!!

Answer 6

here is my proof. the graph is a line. all lines are are one to one except a horizontal line. (which fails the horizontal line test spectacularly)

Answer 7

and of course a vertical line,which does not represent a function