Determine whether the function is one-to-one.
-$$f ( x ) = 4 x ^ { 2 } + x$$

Question

Quizplus · Accepted Answer

To show that a function is not one-to-one, we need to find two different inputs that produce the same output. Let's suppose that $f(a) = f(b)$ for some $a$ and $b$. Then we have 
$$4a^2 + a = 4b^2 + b$$
which can be rearranged as 
$$4a^2 - 4b^2 = b - a.$$
We can factor the left side as $(2a+2b)(2a-2b) = 4(a-b)(a+b)$, and rewrite the equation as 
$$(a+b)(2a-2b) = b-a.$$ 
If $a
eq b$, then $(a+b)$ and $(2a-2b)$ have opposite signs, so their product is negative. But $(b-a)$ is positive, since $b>a$. Therefore, we have a contradiction, and the assumption that $f(a) = f(b)$ must be false. Therefore, $f$ is not one-to-one.

Determine Whether the Function Is One-To-One f(x)=4x2+xf ( x ) = 4 x ^ { 2 } + xf(x)=4x2+x

Determine Whether the Function Is One-To-One $f ( x ) = 4 x ^ { 2 } + x$