Question 1135135

show that the given function is one-to-one and find its inverse. Check your
answers algebraically and graphically. Verify that the range of f is the domain of f^−1 and vice-versa. 

{{{f(x) = 3sqrt(x-1) - 4 }}}

It is a 1-1 function if it passes both the {{{vertical }}}line test and the {{{horizontal}}} line test. 

if {{{f(a) =f(b) }}} implies that a=b, then {{{f(x)}}} is 1-1

{{{3sqrt(a-1) - 4= 3sqrt(b-1) - 4}}}

{{{3sqrt(a-1) = 3sqrt(b-1)}}}

{{{sqrt(a-1) = sqrt(b-1)}}}

{{{a-1 = b-1}}}
{{{a = b}}}

so, your function is injective (one-to-one)

inverse:

{{{f(x) = 3sqrt(x-1) - 4 }}}......{{{f(x) = y }}}

{{{y= 3sqrt(x-1) - 4 }}}......swap {{{x}}} and {{{y}}}

{{{x= 3sqrt(y-1) - 4 }}}.......solve for {{{y}}}

{{{x+4= 3sqrt(y-1)  }}}

{{{(x+4)/3= sqrt(y-1)  }}}.........square both sides


{{{((x+4)/3)^2= (sqrt(y-1))^2  }}}

{{{(x+4)^2/9= y-1  }}}

{{{y=x^2/9+8x/9+16/9+1  }}}

{{{y=x^2/9+8x/9+16/9+9/9  }}}

{{{y=x^2/9+8x/9+25/9  }}}

{{{y=(1/9)(x^2+8x+25)  }}}


{{{f^-1(x)=(1/9)(x^2+8x+25) }}}


{{{ graph( 600, 600, -5, 10, -10, 10, 3sqrt(x-1) - 4, (1/9)(x^2+8x+25)) }}}