To find the function [tex]\(g(x)\)[/tex], we start with the given functions:
1. [tex]\(f(x) = 9x + 9\)[/tex]
2. [tex]\((f+g)(x) = 10 - \frac{1}{3}x\)[/tex]
We know that [tex]\((f + g)(x)\)[/tex] is the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. Therefore, we can write:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substituting the given expressions for [tex]\(f(x)\)[/tex] and [tex]\((f+g)(x)\)[/tex], we get:
[tex]\[ 10 - \frac{1}{3}x = 9x + 9 + g(x) \][/tex]
Next, we need to isolate [tex]\(g(x)\)[/tex]. To do this, we will subtract [tex]\(9x + 9\)[/tex] from both sides of the equation:
[tex]\[ g(x) = 10 - \frac{1}{3}x - (9x + 9) \][/tex]
Now, simplify the right-hand side of the equation step-by-step:
1. Distribute the negative sign:
[tex]\[ g(x) = 10 - \frac{1}{3}x - 9x - 9 \][/tex]
2. Combine like terms. Notice that [tex]\(10\)[/tex] and [tex]\(-9\)[/tex] are constants, and [tex]\(-\frac{1}{3}x\)[/tex] and [tex]\(-9x\)[/tex] are terms with [tex]\(x\)[/tex]:
[tex]\[ g(x) = (10 - 9) - \left(\frac{1}{3}x + 9x\right) \][/tex]
3. Simplify [tex]\(10 - 9\)[/tex]:
[tex]\[ g(x) = 1 - \left(\frac{1}{3}x + 9x\right) \][/tex]
4. To combine [tex]\(\frac{1}{3}x\)[/tex] and [tex]\(9x\)[/tex], express [tex]\(9x\)[/tex] as a fraction with a common denominator. [tex]\(9 = \frac{27}{3}\)[/tex]:
[tex]\[ g(x) = 1 - \left(\frac{1}{3}x + \frac{27}{3}x\right) \][/tex]
5. Combine the fractions:
[tex]\[ g(x) = 1 - \frac{28}{3}x \][/tex]
Putting it all together, we find:
[tex]\[ g(x) = 1 - \frac{28}{3}x \][/tex]
So, the function [tex]\(g(x)\)[/tex] is:
[tex]\[ g(x) = 1 - \frac{28}{3}x \][/tex]
