Certainly! Let's find the function [tex]\( g(x) \)[/tex] given the following functions:
[tex]\[ f(x) = 2x^2 \][/tex]
[tex]\[ a(x) = \left( -\frac{1}{2} \right) x^2 \][/tex]
Our goal is to determine [tex]\( g(x) \)[/tex], which is the difference between [tex]\( f(x) \)[/tex] and [tex]\( a(x) \)[/tex]. Mathematically, we express this as:
[tex]\[ g(x) = f(x) - a(x) \][/tex]
Substituting the given functions [tex]\( f(x) \)[/tex] and [tex]\( a(x) \)[/tex] into this expression, we have:
[tex]\[ g(x) = 2x^2 - \left( -\frac{1}{2} \right)x^2 \][/tex]
Now, simplify the expression inside the parentheses:
[tex]\[ g(x) = 2x^2 + \frac{1}{2}x^2 \][/tex]
Next, combine the like terms [tex]\( 2x^2 \)[/tex] and [tex]\( \frac{1}{2}x^2 \)[/tex]:
To do this, make the denominators the same:
[tex]\[ 2x^2 = \frac{4}{2}x^2 \][/tex]
So,
[tex]\[ g(x) = \frac{4}{2}x^2 + \frac{1}{2}x^2 \][/tex]
[tex]\[ g(x) = \left( \frac{4}{2} + \frac{1}{2} \right)x^2 \][/tex]
[tex]\[ g(x) = \frac{5}{2}x^2 \][/tex]
Thus, the function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \frac{5}{2}x^2 \][/tex]
There you have it, the function [tex]\( g(x) \)[/tex] based on your new definitions is:
[tex]\[ g(x) = \frac{5}{2}x^2 \][/tex]
