Question
The function f(x)=0.04x represents the amount Sally earns in commission where x is the amount of sales in dollars. The function g(x)=x−4,000 represents the amount of sales on which Sally gets paid commission. What is (f∘g)(x) and what does it mean? What is the domain?(1 point)Responses(f∘g)(x)=0.04x−160 represents the amount Sally earns in commission. The domain is x≥0.open paren f circ g close paren times x is equal to 0 point 0 4 x minus 160 represents the amount Sally earns in commission. The domain is x is greater than or equal to 0 .(f∘g)(x)=0.04x−160 represents the amount Sally earns in commission. The domain is (−∞, ∞). open paren f circ g close paren times x is equal to 0 point 0 4 x minus 160 represents the amount Sally earns in commission. The domain is open paren negative infinity comma infinity close paren .(f∘g)(x)=0.04x−4,000 represents the amount Sally earns in salary and commission. The domain is x≥0.open paren f circ g close paren times x is equal to 0 point 0 4 x minus 4 comma 000 represents the amount Sally earns in salary and commission. The domain is x is greater than or equal to 0 .(f∘g)(x)=0.04x−4,000 represents the amount Sally earns i
Answers
GPT-4o mini
To find \((f \circ g)(x)\), we need to compose the two functions \(f(x) = 0.04x\) and \(g(x) = x - 4,000\). This means we will take \(g(x)\) and substitute it into \(f(x)\).
1. Start with \(g(x)\):
\[
g(x) = x - 4,000
\]
2. Now substitute \(g(x)\) into \(f(x)\):
\[
(f \circ g)(x) = f(g(x)) = f(x - 4,000)
\]
\[
= 0.04(x - 4,000)
\]
3. Distribute \(0.04\):
\[
= 0.04x - 0.04 \times 4,000
\]
\[
= 0.04x - 160
\]
Thus, \((f \circ g)(x) = 0.04x - 160\).
### Interpretation
This expression represents the amount Sally earns in commission based on her total sales \(x\) after subtracting the threshold of $4,000 (the amount of sales on which she gets paid commission). Specifically, she earns a 4% commission (0.04 factor) on sales above $4,000, but she automatically starts with a deduction of $160.
### Domain
Given that \(g(x) = x - 4,000\) affects the commission, we need to consider the conditions under which Sally receives commission (i.e., when \(g(x) \geq 0\)):
\[
x - 4,000 \geq 0
\]
\[
x \geq 4,000
\]
So the domain of \( (f \circ g)(x) \) is \( x \geq 4,000 \).
### Conclusion
The final result is:
- \((f \circ g)(x) = 0.04x - 160\) represents the amount Sally earns in commission.
- The domain is \( x \geq 4,000 \).
1. Start with \(g(x)\):
\[
g(x) = x - 4,000
\]
2. Now substitute \(g(x)\) into \(f(x)\):
\[
(f \circ g)(x) = f(g(x)) = f(x - 4,000)
\]
\[
= 0.04(x - 4,000)
\]
3. Distribute \(0.04\):
\[
= 0.04x - 0.04 \times 4,000
\]
\[
= 0.04x - 160
\]
Thus, \((f \circ g)(x) = 0.04x - 160\).
### Interpretation
This expression represents the amount Sally earns in commission based on her total sales \(x\) after subtracting the threshold of $4,000 (the amount of sales on which she gets paid commission). Specifically, she earns a 4% commission (0.04 factor) on sales above $4,000, but she automatically starts with a deduction of $160.
### Domain
Given that \(g(x) = x - 4,000\) affects the commission, we need to consider the conditions under which Sally receives commission (i.e., when \(g(x) \geq 0\)):
\[
x - 4,000 \geq 0
\]
\[
x \geq 4,000
\]
So the domain of \( (f \circ g)(x) \) is \( x \geq 4,000 \).
### Conclusion
The final result is:
- \((f \circ g)(x) = 0.04x - 160\) represents the amount Sally earns in commission.
- The domain is \( x \geq 4,000 \).