Question

how would i figure out if f(x) = x4 + 2; 0 ≤ x ≤ 6
is a "one to one" function?

Answer 1

you could find the inverse, if the inverse if a function over the domain. then f(x) is one-to-one

Answer 2

the domain being 0 ≤ x ≤ 6 ?

Answer 3

yes

Answer 4

if f(x) outputs only 1 value for each x in the domain, then f(x) is one-to-one

Answer 5

normally f(x)=x^4+2 would not be one-to-one, because f(x)=f(-x) for all x in the domain of real numbers. if you exclude the negative numbers, f(x) is one-to-one

Answer 6

can u explain 0 ≤ x ≤ 6 to me? i dont think i understand how to read that

Answer 7

does that mean between 0 and 6?

Answer 8

less than or equal to
interval notation would be [0,6]

Answer 9

so the answer could only be between 0 and 6 INclusive?

Answer 10

no, the inputs could only be between 0 and 6.
the outputs would be the results of these inputs

Answer 11

okay ill have to maul this over in my brain a bit.. thank you though

Answer 12

yw

Answer 13

i have an idea, find the vertex and then see whether the function is strictly increasing or decreasing over your interval, i.e. find if the vertex is in the inteval or not

Answer 14

how would i go about doing that?

Answer 15

since the vertex of \(y=x^4+2\) is the point (0,2) then over the interval \((0,\infty)\) your function is increasing

Answer 16

and hence is one to one

Answer 17

we can reason as follows: \(x^4\geq 0\) for any value of x, and so it has a minimum value at x = 0
at x = 0
the function will be decreasing on \((-\infty,0)\) and increasing on \((0,\infty)\)
if the function is increasing then it is one to one, so since it is increasing on \((0,\infty)\) it is also increasing on \((0,6)\)

Answer 18

so only if the function is is increasing is it one to one? and how does the given domain come into play?

Answer 19

yes if it is strictly increasing (or decreasing) it is one to one

Answer 20

if it is increasing this means \(f(a)<f(b)\) provided that \(a<b\) and one to one means
\[f(a)=f(b)\implies a=b\] so the first statement implies the second, that is, if to inputs are different, the outputs must be different as well

Answer 21

*two inputs

Answer 22

this is just so complicated to me.. what is the simplist way for me to answer the question of whether it is or isnt a one to one function of an exam

Answer 23

on a exam*

Answer 24

how did you know that the vertix was (0,2)?