Question

What is f(g(x)) for x > 5?

Answer 1

ANSWER

$f(g(x)) =4{x}^{2} - 41x + 105$

EXPLANATION

The given functions are:

$f(x) = 4x - \sqrt{x}$

and

$g(x) = {(x - 5)}^{2}$

To find

$f(g(x)) = f( {(x - 5)}^{2} )$

$f(g(x)) =4 {(x - 5)}^{2} - \sqrt{{(x - 5)}^{2} }$

We expand and simplify to obtain,

$f(g(x)) =4 {( {x}^{2} - 10x + 25)} - (x - 5)$

$f(g(x)) =4{x}^{2} - 40x + 100 - x + 5$

Combine similar terms to get;

$f(g(x)) =4{x}^{2} - 41x + 105$

The correct choice is B.

Answer 2

f(g(x))  (you plug/substitute g(x) into x)

f(x) = 4x - √x

f(g(x)) = 4(g(x)) - √(g(x))      since g(x) = (x - 5)², you can do:

f(g(x)) = 4(x - 5)² - √(x - 5)²     The ² and √ cancel each other, leaving

f(g(x)) = 4(x - 5)² - (x - 5)      Next factor out (x - 5)² or (x - 5)(x - 5)

f(g(x)) = 4(x² - 10x + 25) - (x - 5)  Now distribute the 4 and the -

f(g(x)) = 4x² - 40x + 100 - x + 5   Simplify

f(g(x)) = 4x² - 41x + 105         Your answer is B