Sure, let's proceed by explaining the piecewise function step-by-step to understand Kathryn's rates for different ranges of [tex]\( x \)[/tex], where [tex]\( x \)[/tex] represents the number of hours of consulting.
1. First Interval: [tex]\( 0 \leq x \leq 4 \)[/tex]
For this interval, [tex]\( f(x) = 50x \)[/tex].
This means that if Kathryn works between 0 and 4 hours inclusive, she charges a rate of \[tex]$50 per hour. So:
- For 1 hour, she charges \( 50 \cdot 1 = \$[/tex]50 \).
- For 2 hours, she charges [tex]\( 50 \cdot 2 = \$100 \)[/tex].
- For 3 hours, she charges [tex]\( 50 \cdot 3 = \$150 \)[/tex].
- For 4 hours, she charges [tex]\( 50 \cdot 4 = \$200 \)[/tex].
2. Second Interval: [tex]\( 4 < x \leq 8 \)[/tex]
For this interval, [tex]\( f(x) = 40x \)[/tex].
This means that if Kathryn works more than 4 hours and up to 8 hours, she charges a rate of \[tex]$40 per hour. Starting from 4 hours, the calculation switch:
- If she worked exactly 5 hours, she would charge \( 40 \cdot 5 = \$[/tex]200 \).
- For 6 hours, she charges [tex]\( 40 \cdot 6 = \$240 \)[/tex].
- For 7 hours, she charges [tex]\( 40 \cdot 7 = \$280 \)[/tex].
- For 8 hours, she charges [tex]\( 40 \cdot 8 = \$320 \)[/tex].
3. Third Interval: [tex]\( x > 8 \)[/tex]
For this interval, [tex]\( f(x) = 25x \)[/tex].
This means that if Kathryn works more than 8 hours, she charges a rate of \[tex]$25 per hour. Starting from any time beyond 8 hours:
- For 9 hours, she charges \( 25 \cdot 9 = \$[/tex]225 \).
- For 10 hours, she charges [tex]\( 25 \cdot 10 = \$250 \)[/tex].
- For 12 hours, she charges [tex]\( 25 \cdot 12 = \$300 \)[/tex].
In summary, Kathryn's rates are determined by the intervals in which the number of consulting hours [tex]\( x \)[/tex] falls:
- Between 0 and 4 hours, she charges [tex]\( 50x \)[/tex].
- From greater than 4 up to 8 hours, she charges [tex]\( 40x \)[/tex].
- More than 8 hours, she charges [tex]\( 25x \)[/tex].
