Answer
Only $\text{(a) }h(x)=2x$ has the given property.
Work Step by Step
Check each function if the property applies:
a)
$h(x)=2x$
$h(a+b)=2(a+b)=2a+2b=h(a)+h(b)$
$h(a+b)=h(a)+h(b)$
Therefore the function $h(x)=2x$ has the property for any real numbers $a$ and $b$
b)
$g(x)=x^2$
$g(a+b)=(a+b)^2=a^2+2ab+b^2$
$g(a)+g(b) = a^2+b^2 $
$g(a+b)\ne g(a)+g(b)$
Therefore the function $g(x)=x^2$ does not have the property for any real numbers $a$ and $b$
c)
$F(x)=5x-2$
$F(a+b)=5(a+b)-2=5a+5b-2$
$F(a)+F(b) = 5a-2+5b-2 =5a+5b-4 $
$F(a+b) \ne F(a)+F(b)$
Therefore the function $F(x)=5x-2$ does not have the property for any real numbers $a$ and $b$
d)
$G(x)= \frac{1}{x}$
$G(a+b) = \frac{1}{a+b} $
$G(a)+G(b) = \frac{1}{a}+\frac{1}{b} $
$G(a+b) \ne G(a)+G(b) $
Therefore the function $G(x)= \frac{1}{x}$ does not have the property for any real numbers $a$ and $b$
