Let's start by deriving the combined income function [tex]\( h(x) \)[/tex].
1. Determine the income functions:
- Colby's income function is [tex]\( f(x) = 3x + 12 \)[/tex].
- Danielle's income function is [tex]\( g(x) = 5x + 10 \)[/tex].
2. Combine these functions to find [tex]\( h(x) \)[/tex], their combined income:
- [tex]\( h(x) = f(x) + g(x) \)[/tex].
- Therefore, [tex]\( h(x) = (3x + 12) + (5x + 10) \)[/tex].
- Simplifying, [tex]\( h(x) = 3x + 5x + 12 + 10 \)[/tex].
- Thus, [tex]\( h(x) = 8x + 22 \)[/tex].
So, the function [tex]\( h(x) \)[/tex] is [tex]\( h(x) = 8x + 22 \)[/tex].
3. Calculate Colby's income and the combined income for 4 hours:
- Colby's income working alone for 4 hours is [tex]\( f(4) \)[/tex].
- [tex]\( f(4) = 3(4) + 12 \)[/tex].
- [tex]\( f(4) = 12 + 12 \)[/tex].
- [tex]\( f(4) = 24 \)[/tex].
- The combined income if Colby and Danielle work together for 4 hours is [tex]\( h(4) \)[/tex].
- [tex]\( h(4) = 8(4) + 22 \)[/tex].
- [tex]\( h(4) = 32 + 22 \)[/tex].
- [tex]\( h(4) = 54 \)[/tex].
4. Compare the incomes:
- Colby's income alone is [tex]\( 24 \)[/tex].
- The combined income is [tex]\( 54 \)[/tex].
This means Colby will make more money by teaming with Danielle, as the combined income ([tex]\( 54 \)[/tex]) is greater than Colby's individual income ([tex]\( 24 \)[/tex]).
Thus, the correct option is:
[tex]\[ h(x) = 8x + 22, \ \text{team with Danielle} \][/tex]
