Can you please help me find the functions?

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Answer 1 · 2018-05-14T18:33:43

$(g + k) (x) = g (x) + k (x) = x - 3 + 2 x = 3 x - 3$ $(g+k)(x)=g(x)+k(x) = x-3 + 2x = 3x-3$

$(h k) (x) = h (x) k (x) = (x^{2} - 9) (2 x) = 2 x^{3} - 18 x$ $(hk)(x) = h(x)k(x) = (x^2-9)(2x) = 2x^3 - 18x$

$h (g (x)) = h (x - 3) = {(x - 3)}^{2} - 9 = x^{2} - 6 x$ $h(g(x)) = h( x-3) = (x-3)^2 -9 = x^2 - 6x$

$h (g (k (x))) = h (g (2 x)) = {(2 x)}^{2} - 6 (2 x) = 4 x^{2} - 12 x$ $h(g(k(x))) = h(g(2x)) = (2x)^2 - 6(2x) = 4x^2 - 12 x$

Answer 2 · 2018-05-14T18:37:31

$f (x) = x$ $f(x)=x$
$g (x) = x - 3$ $g(x)=x-3$
$h (x) = x^{2} - 9$ $h(x)=x^2-9$
$k (x) = 2 \cdot x$ $k(x)=2*x$

1.
$(g + k) (x) = g (x) + k (x) =$ $(g+k)(x)=g(x)+k(x)=$

$= x - 3 + 2 \cdot x = 3 \cdot (x - 1)$ $=x-3+2*x=3*(x-1)$

2.
$(h \cdot k) (x) = h (x) \cdot k (x)$ $(h*k)(x)=h(x)*k(x)$

$= (x^{2} - 9) \cdot (2 \cdot x) = 2 \cdot x^{3} - 18 x$ $=(x^2-9)*(2*x)=2*x^3-18x$

3.
$h (g (x)) = {(g (x))}^{2} - 9 = {(x - 3)}^{2} - 9 =$ $h(g(x))=(g(x))^2-9=(x-3)^2-9=$

$= x^{2} - 2 \cdot 3 \cdot x + 9 - 9 = x^{2} - 6 \cdot x$ $=x^2-2*3*x+9-9=x^2-6*x$

4.
We can use the) result of 3. replacing $x$ $x$ with $k (x)$ $k(x)$

$h (g (k (x))) = {(k (x))}^{2} - 6 \cdot k (x) = {(2 \cdot x)}^{2} - 6 \cdot 2 \cdot x =$ $h(g(k(x)))=(k(x))^2-6*k(x)=(2*x)^2-6*2*x=$

$= 4 \cdot x^{2} - 12 x$ $=4*x^2-12x$