Do you know your functions?

Algebra Level 4

The following functions are defined as $f(x) : \mathbb{R} \rightarrow \mathbb{R}$

Which of these functions are bijective ?

$1.f(x) = x^2 + 2x - 1$

$2.f(x) = \log(x) + |x|$

$3. f(x)=\log(x)$

$4. f(x)= x^2$

$5. f(x)=2^x$

Assume $\log(x)$ indicates the natural logarithm function.

Only 1 1,2,3,4 and 5 1,4 and 5 2,3,4 and 5 4 and 5 2 and 3 None 1,2 and 3

1 solution

Vishnu Bhagyanath
Jun 24, 2015

A function is bijective if it is both injective and surjective . Hence, disproving any one is enough to prove it is not bijective.

Range is $f(x) \geq -2$
Logarithm of a negative number is not defined. (Or rather, it doesn't come in the set of real numbers)
Same as above.
$x^2 = (-x)^2$ so it is not injective.
Range is $f(x) > 0$