2. means that for every x, there is eactly one y
basically, we can use the vertical line test to see if it crosses at any 2 points
first one works
2nd one works
3rd works
4th works
5th works
they are all one to one functions
3.
easy
steps
for f(x)=?
replace f(x) with y
switch x and y
solve for y
replace y with f⁻¹(x)
so
[tex]f(x)=(1+x)^3-2[/tex]
replace
[tex]y=(1+x)^3-2[/tex]
switch
[tex]x=(1+y)^3-2[/tex]
solve
[tex]x+2=(1+y)^3[/tex]
[tex]\sqrt[3]{x+2}=1+y[/tex]
[tex]-1+\sqrt[3]{x+2}=[/tex]
replace
[tex]f^{-1}(x)=-1+\sqrt[3]{x+2}[/tex]
first option
4.
domain, hmm we can't divide by 0 so domain is all real numbers except for x=0
domain is [tex](-\infty,0) \cup(0,\infty)[/tex]
hmm, range
as x approaches negative or positive infinity, f(x)=5
as x approaches 0 from the left, f(x) appraoches negative infintiy
as x approaches 0 from the right , f(x) appraoches positive infintiy
so range is [tex](-\infty,5) \cup(5,\infty)[/tex]
domain is [tex](-\infty,0) \cup(0,\infty)[/tex]
range is [tex](-\infty,5) \cup(5,\infty)[/tex]
answer is the 4th option
