Digital SAT Maths -Unit 1 - 1.8 Piecewise Functions- Study Notes- New Syllabus
Digital SAT Maths -Unit 1 – 1.8 Piecewise Functions- Study Notes- New syllabus
Digital SAT Maths -Unit 1 – 1.8 Piecewise Functions- Study Notes – per latest Syllabus.
Key Concepts:
Reading piecewise definitions
Evaluating intervals
Reading Piecewise Definitions
A piecewise function is a function defined by different formulas over different intervals of \( x \).
In other words, the rule changes depending on the input value.
It is written in the form:
\( f(x)= \begin{cases} \text{rule 1}, & \text{condition 1}\\ \text{rule 2}, & \text{condition 2}\\ \text{rule 3}, & \text{condition 3} \end{cases} \)
How to Read It
You do not use all formulas. You choose the formula whose interval contains the given \( x \)-value.
Important SAT Idea
The most common mistake is using the wrong interval. Always check the inequality sign carefully.
- \( x < 2 \) → values less than 2 only
- \( x \le 2 \) → includes 2
- \( x > 2 \) → values greater than 2
- \( x \ge 2 \) → includes 2
Endpoint Rule
If two intervals meet at a number (for example 3), only the interval with \( \le \) or \( \ge \) contains that value.
Example 1:
\( f(x)= \begin{cases} 2x+1, & x<0\\ x^2, & x\ge0 \end{cases} \)
Which rule is used when \( x=-3 \)?
▶️ Answer/Explanation
Since \( -3<0 \), use the first rule.
Answer: \( 2x+1 \).
Example 2 (Endpoint):
\( g(x)= \begin{cases} x+4, & x\le3\\ 5x, & x>3 \end{cases} \)
Which rule applies when \( x=3 \)?
▶️ Answer/Explanation
The first interval includes 3 because of \( \le \).
Answer: \( x+4 \).
Example 3 (SAT Context):
A parking garage charges
\( C(t)= \begin{cases} 5, & 0<t\le2\\ 5+3(t-2), & t>2 \end{cases} \)
Which rule applies for 1.5 hours?
▶️ Answer/Explanation
Since \( 1.5 \le 2 \), use the first rule.
Answer: Cost = 5 dollars rule.
Evaluating Intervals
After identifying the correct interval, the next step is to evaluate the function using that rule.
Key idea:
Choose the correct formula first, then substitute the value.
Never substitute into every expression. Only use the one whose condition is true.
Common DIGITAL SAT Trick
The SAT often uses boundary numbers such as 2, 5, or 10 to check if you understand \( < \) versus \( \le \).
Steps
- Locate which interval the input belongs to
- Select the correct expression
- Substitute and simplify
Example 1:
\( f(x)= \begin{cases} x+2, & x<1\\ 3x, & x\ge1 \end{cases} \)
Find \( f(4) \).
▶️ Answer/Explanation
Since \( 4 \ge 1 \), use \( 3x \).
\( f(4)=3(4)=12 \)
Answer: 12
Example 2 (Boundary Value):
\( g(x)= \begin{cases} 2x-1, & x\le5\\ x^2, & x>5 \end{cases} \)
Find \( g(5) \).
▶️ Answer/Explanation
Because of \( \le \), 5 belongs to the first interval.
\( g(5)=2(5)-1=9 \)
Answer: 9
Example 3 (SAT Context):
A delivery company charges
\( C(w)= \begin{cases} 8, & 0<w\le2\\ 8+4(w-2), & w>2 \end{cases} \)
Find the cost for a package weighing 3 kg.
▶️ Answer/Explanation
Since \( 3>2 \), use the second rule.
\( C(3)=8+4(3-2) \)
\( =8+4=12 \)
Conclusion: The delivery cost is \$12.