# Verify that x = 1 => fg(x) = gf(x)

• Jun 27th 2010, 03:43 PM
ansonbound
Verify that x = 1 => fg(x) = gf(x)
a)The functions f and g are defined by f(x) = root x for x > 0; g(x) = x - 1 for all values of x.

i.)Write down expressions for fg(x) and gf(x).
> fg(x) = root x - 1
> gf(x) = root x - 1

ii.) Verify that x = 1 => fg(x) = gf(x)
>??????
• Jun 27th 2010, 04:15 PM
skeeter
Quote:

Originally Posted by ansonbound
a)The functions f and g are defined by f(x) = root x for x > 0; g(x) = x - 1 for all values of x.

i.)Write down expressions for fg(x) and gf(x).
> fg(x) = root x - 1
> gf(x) = root x - 1

ii.) Verify that x = 1 => fg(x) = gf(x)
>??????

$\displaystyle f(x) = \sqrt{x}$ , $\displaystyle x > 0$

$\displaystyle g(x) = x - 1$

I assume you mean function composition by the notation "fg(x) and gf(x)"

$\displaystyle f[g(x)] = f(x-1) = \sqrt{x-1}$

$\displaystyle g[f(x)] = g(\sqrt{x}) = \sqrt{x} - 1$

you should be able to see that $\displaystyle f[g(1)] = g[f(1)] = 0$
• Jun 27th 2010, 04:20 PM
ansonbound
Quote:

Originally Posted by skeeter
$\displaystyle f(x) = \sqrt{x}$ , $\displaystyle x > 0$

$\displaystyle g(x) = x - 1$

I assume you mean function composition by the notation "fg(x) and gf(x)"

$\displaystyle f[g(x)] = f(x-1) = \sqrt{x-1}$

$\displaystyle g[f(x)] = g(\sqrt{x}) = \sqrt{x} - 1$

you should be able to see that $\displaystyle f[g(1)] = g[f(1)] = 0$

so it is right that i say $\displaystyle \sqrt{x-1} = \sqrt{x} - 1 = 0?$
• Jun 27th 2010, 04:30 PM
skeeter
Quote:

Originally Posted by ansonbound
so it is right that i say $\displaystyle \sqrt{x-1} = \sqrt{x} - 1 = 0?$

not for all x ... only for x = 1
• Jun 27th 2010, 04:58 PM
ansonbound
• Jun 27th 2010, 05:04 PM
skeeter
$\displaystyle y = \sqrt{x-1}$

swap variables ...

$\displaystyle x = \sqrt{y-1}$

solve for y to finish
• Jun 27th 2010, 05:53 PM
ansonbound
• Jun 27th 2010, 06:02 PM
ansonbound
Quote:
yes! i did it by using
$\displaystyle b^2 - 4ac = 0 16 - 4*k*2 = 16 k = 2$
• Jun 27th 2010, 06:05 PM
skeeter
sketch the graph y = |2x-4|

sketch the graph y = x

how far up will you have to translate y = |2x-4| so that it touches y = x at only one point?