Piecewise Functions UPCAT Reviewer: Lesson and Practice

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Piecewise Functions

Choose the interval before calculating; the condition determines which rule is allowed.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Piecewise Functions

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Choose the condition first; only then use its formula

A piecewise function assigns different formulas to different parts of its domain. For example:

f(x) = x+3   if x<2f(x) = 2x−1   if x≥2

The conditions split the number line at x=2. An input below 2 uses the first formula. An input equal to or above 2 uses the second. The equality sign tells you which branch owns the boundary.

1. Compare the input

Check the input against every condition before calculating.

2. Use one branch only

Once the condition matches, ignore the other formulas.

DO IT FAST

Circle the input, compare it with every condition

Problem: f(x)=x+4 for x<2 and 2x−1 for x≥2. Find f(2).

Recognize: 2 does not satisfy x<2, but it does satisfy x≥2.

Do it fast: Use 2x−1: f(2)=4−1=3.

Why it works

Most errors come from calculating with both formulas or overlooking which inequality contains the equality sign.

WORKED EXAMPLES

Five forms you should recognize

1. Input below the boundary

f(x)={ x+3 if x<2; 2x−1 if x≥2 }

Find f(1). Since 1<2, use x+3:

f(1)=1+3=4Answer: 4

2. Input exactly at the boundary

g(x)={ x+4 if x<3; 2x−1 if x≥3 }

Find g(3). The first condition excludes 3, while x≥3 includes it:

g(3)=2(3)−1=5Answer: 5

3. Open and closed graph endpoints

At x=2, the point at y=4 is open, so it is excluded. The point at y=1 is filled, so it gives the function value.

Therefore f(2)=1.

4. Tiered parking charge

A parking lot charges ₱40 for the first 2 hours and ₱25 for every additional hour.

C(h)={ 40 if 0<h≤2; 40+25(h−2) if h>2 }

For 5 hours, three hours are beyond the first two:

C(5)=40+25(5−2)=40+75=₱115

Do not charge all five hours at ₱25; the first two are already included in ₱40.

5. Making two branches meet

p(x)={ 2x+k if x<3; x²−1 if x≥3 }

To make the graph meet at x=3, the branch values at the boundary must agree. The second branch gives:

3²−1=8

Set the first expression equal to 8:

2(3)+k=8 → 6+k=8 → k=2The branches meet when k=2.

COMMON TRAPS

Check before you commit

• Substituting before checking the condition
• Using both formulas for one input
• Ignoring which inequality contains the equality sign
• Treating an open graph point as included
• Charging every unit at the higher tier instead of only the excess
• Assuming every piecewise graph has a jump

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

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A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

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Piecewise Functions FAQ

Why are there two mastery sets?

An unsuccessful attempt loads a different five-question set so memorizing one set does not verify mastery.

Does 5/5 mean permanent mastery?

It verifies the competency today. Revisit it later through mixed practice and simulations.

RELATED COMPETENCIES

Continue your mathematics review.

FunctionsDomain and RangeLinear Functions

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