Digital SAT Maths -Unit 1 - 1.6 Function its Notation and Composite Function- Study Notes- New Syllabus
Digital SAT Maths -Unit 1 – 1.6 Function its Notation and Composite Function- Study Notes- New syllabus
Digital SAT Maths -Unit 1 – 1.6 Function its Notation and Composite Function- Study Notes – per latest Syllabus.
Key Concepts:
Functions, Function Notation, Domain and Range
Evaluating \( f(x) \), \( g(x) \) and Interpreting Meaning in Context
Composite Function
Functions, Function Notation, Domain and Range
A function is a rule that assigns exactly one output to each input.
Each value of \( x \) produces only one value of \( y \).
On the DIGITAL SAT, you are commonly asked to evaluate functions, interpret meaning, and identify the domain and range.
Function Notation
Instead of writing
\( y = 2x + 3 \)
we write
\( f(x) = 2x + 3 \)
- \( x \) → input
- \( f(x) \) → output
Evaluating a Function
Substitute the value into the function.
Domain
The domain is the set of all possible input values (\( x \)-values).
For most linear functions:
Domain = all real numbers
Range
The range is the set of all possible output values (\( y \)-values).
For non-horizontal linear functions:
Range = all real numbers
Function vs Not a Function
A relation is a function only if each input has one output.
- Function: \( (1,4), (2,6), (3,8) \)
- Not a function: \( (1,4), (2,6), (1,9) \)
Vertical Line Test
A graph represents a function if a vertical line touches the graph at most once.
DIGITAL SAT Meaning
If a function models a situation:
- Domain → allowed inputs (time, quantity, people)
- Range → possible results (cost, height, distance)
Example 1 (Evaluate & Domain):
Given \( f(x)=2x+5 \), find \( f(6) \) and state the domain.
▶️ Answer/Explanation
\( f(6)=2(6)+5=17 \)
Linear functions accept all real \( x \).
Domain: all real numbers.
Example 2 (Range in Context):
A taxi fare is modeled by \( C(t)=4t+10 \), where \( t \ge 0 \) hours. What is the domain and range?
▶️ Answer/Explanation
Domain: \( t \ge 0 \)
Minimum cost occurs at \( t=0 \).
\( C(0)=10 \)
Range: \( C \ge 10 \)
Example 3 (Is it a Function?):
Determine whether the relation is a function and list the range:
\( (2,5), (4,7), (6,5), (8,9) \)
▶️ Answer/Explanation
Each input appears once → it is a function.
Range = set of outputs:
\( \{5,7,9\} \)
Conclusion: Function with range \( \{5,7,9\} \).
Evaluating \( f(x) \), \( g(x) \) and Interpreting Meaning in Context
On the DIGITAL SAT, you will often see more than one function, such as \( f(x) \) and \( g(x) \). You may be asked to evaluate them, compare them, or explain what a value means in a real-world situation.
Evaluating a Function
To evaluate a function, replace the input with the given number.
If \( f(x)=3x+2 \), then \( f(5)=3(5)+2=17 \).
Evaluating Two Functions
Each function is separate. Always substitute into the correct one.
\( f(2) \) means use the rule for \( f \)
\( g(2) \) means use the rule for \( g \)
Operations with Functions
- \( f(a)+g(a) \) → evaluate each, then add
- \( f(a)-g(a) \) → evaluate each, then subtract
Interpreting in Context
In word problems, function notation represents a real quantity.
- \( C(t) \) → cost after \( t \) months
- \( d(t) \) → distance after time \( t \)
- \( P(x) \) → profit after selling \( x \) items
Important SAT Idea
The test often does NOT want the number alone. It wants what the number represents.
Example 1 (Evaluate Two Functions):
Given
\( f(x)=2x+1 \)
\( g(x)=x^2-3 \)
Find \( f(3) \) and \( g(3) \).
▶️ Answer/Explanation
\( f(3)=2(3)+1=7 \)
\( g(3)=3^2-3=9-3=6 \)
Conclusion: \( f(3)=7 \), \( g(3)=6 \).
Example 2 (Function Operations):
Using the same functions, find \( f(2)+g(2) \).
▶️ Answer/Explanation
\( f(2)=2(2)+1=5 \)
\( g(2)=2^2-3=4-3=1 \)
\( f(2)+g(2)=6 \)
Conclusion: The value is 6.
Example 3 (Meaning in Context):
A streaming service charges according to \( C(t)=8t+5 \), where \( t \) is months. What does \( C(4) \) represent?
▶️ Answer/Explanation
\( C(4)=8(4)+5=37 \)
Conclusion: The total cost after 4 months is \$37.
Composite Functions
A composite function is formed when the output of one function becomes the input of another function.
In notation, this is written as:
\( (f \circ g)(x) \)
and is read as:
“\( f \) of \( g \) of \( x \)”
Important Meaning
\( (f \circ g)(x) \) means:
First apply \( g \), then apply \( f \).
How to Evaluate a Composite Function
- Find \( g(x) \)
- Substitute that expression into \( f \)
- Simplify
Order Matters
In general:
\( (f \circ g)(x) \ne (g \circ f)(x) \)
DIGITAL SAT Tip
The SAT often tests whether you understand the order of substitution, not just algebra.
Example 1 (Find a Composite):
Given
\( f(x)=2x+1 \)
\( g(x)=x-3 \)
Find \( (f \circ g)(x) \).
▶️ Answer/Explanation
Start with \( g(x) \):
\( g(x)=x-3 \)
Substitute into \( f \):
\( f(g(x))=2(x-3)+1 \)
\( =2x-6+1 \)
\( =2x-5 \)
Conclusion: \( (f \circ g)(x)=2x-5 \).
Example 2 (Evaluate a Composite):
Using the same functions, find \( (f \circ g)(4) \).
▶️ Answer/Explanation
First find \( g(4) \):
\( g(4)=4-3=1 \)
Now apply \( f \):
\( f(1)=2(1)+1=3 \)
Conclusion: \( (f \circ g)(4)=3 \).
Example 3 (Context Interpretation):
A factory models production by \( g(t)=5t \), where \( t \) is hours, and profit by \( f(x)=2x-40 \), where \( x \) is units produced. What does \( (f \circ g)(t) \) represent?
▶️ Answer/Explanation
First find the composite:
\( (f \circ g)(t)=f(5t)=2(5t)-40 \)
\( =10t-40 \)
Conclusion: \( (f \circ g)(t) \) represents the profit after \( t \) hours of production.