Master function notation, evaluation, and transformations for SAT
How can I study Functions effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 16 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Functions study guide free?▾
Yes — all study notes, flashcards, and practice problems for Functions on Study Mondo are 100% free. No account is needed to access the content.
What course covers Functions?▾
Functions is part of the SAT Prep course on Study Mondo, specifically in the Heart of Algebra section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Functions?
5
)
=
2(5)+
3=
13
Types of SAT Function Questions
1. Evaluation
"If f(x)=x2−3, what is f(4)?"
Answer: f(4)=16−3=13
2. Composition
(f∘g)(x)=f(g(x))
If f(x)=2x and g(x)=x+1:
f(g(3))=f(4)=8
3. Finding Input
"If f(x)=3x−1 and f(a)=14, what is a?"
Solve: 3a−1=14⇒a=5
Domain and Range
Domain: All possible input values (x-values)
Range: All possible output values (y-values)
SAT Function Tricks
Watch for:
f(x+2) vs f(x)+2 (very different!)
Questions asking for 2f(3) when they give you f(3)=5
Answer: 2(5)=10, not f(6)!
f(6)
💡 Show Solution
Solution:
Substitute x=6:
f(6)=3(6)−4f(6)=18−4f(6)=14
Answer:14
2Problem 2easy
❓ Question:
If f(x)=3x−4, what is f(6)?
💡 Show Solution
Solution:
Substitute x=6:
f(6)=3(6)−4
3Problem 3medium
❓ Question:
If g(x)=x2+2x, what is g(−3)?
💡 Show Solution
Solution:
Substitute x=−3:
g(−3)=(−3)
4Problem 4medium
❓ Question:
If g(x)=x2+2x, what is g(−3)?
💡 Show Solution
Solution:
Substitute x=−3:
g(−3)=(−3)
5Problem 5hard
❓ Question:
If h(x)=2x+5 and h(a)=17, what is the value of a?
💡 Show Solution
Solution:
Set up the equation:
h(a)=172a+5=17
Solve:
6Problem 6hard
❓ Question:
If h(x)=2x+5 and h(a)=17, what is the value of a?
💡 Show Solution
Solution:
Set up the equation:
h(a)=172a+5=17
Solve:
7Problem 7easy
❓ Question:
If f(x)=3x−7, what is the value of f(5)?
💡 Show Solution
Step 1: Replace every x with 5 in the function:
f(5)=3(5)−7
Step 2: Evaluate:
8Problem 8easy
❓ Question:
If f(x)=3x−7, what is the value of f(5)?
💡 Show Solution
Step 1: Replace every x with 5 in the function:
f(5)=3(5)−7
Step 2: Evaluate:
9Problem 9medium
❓ Question:
If f(x)=x2−4x+3, for what value(s) of x does f(x)=0?
💡 Show Solution
Step 1: Set the function equal to zero:
x2−4x+3=0
Step 2: Factor:
10Problem 10medium
❓ Question:
If f(x)=x2−4x+3, for what value(s) of x does f(x)=0?
💡 Show Solution
Step 1: Set the function equal to zero:
x2−4x+3=0
Step 2: Factor:
11Problem 11medium
❓ Question:
The function g is defined by g(x)=2x+1. If g(a)=13, what is the value of g(2a)?
💡 Show Solution
Step 1: Find a using g(a)=13:
2a+1=
12Problem 12medium
❓ Question:
The function g is defined by g(x)=2x+1. If g(a)=13, what is the value of g(2a)?
💡 Show Solution
Step 1: Find a using g(a)=13:
2a+1=
13Problem 13hard
❓ Question:
If f(x)=2x+3 and g(x)=x2−1, what is f(g(2))?
💡 Show Solution
Step 1: Evaluate the inner function first — find g(2):
g(2)=(2)2−
14Problem 14hard
❓ Question:
If f(x)=2x+3 and g(x)=x2−1, what is f(g(2))?
💡 Show Solution
Step 1: Evaluate the inner function first — find g(2):
g(2)=(2)2−
15Problem 15expert
❓ Question:
The graph of y=f(x) passes through the point (3,7). If g(x)=f(2x−1)+3, what point must lie on the graph of y=g(x)?
💡 Show Solution
Step 1: We know f(3)=7 (since the graph passes through (3,7)).
Step 2: We need to find a value of x where can be evaluated.
For , we need (so that is evaluated at 3, which we know).
16Problem 16expert
❓ Question:
The graph of y=f(x) passes through the point (3,7). If g(x)=f(2x−1)+3, what point must lie on the graph of y=g(x)?
💡 Show Solution
Step 1: We know f(3)=7 (since the graph passes through (3,7)).
Step 2: We need to find a value of x where can be evaluated.
For , we need (so that is evaluated at 3, which we know).
▾
Yes, this page includes 16 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.
f(6)=
18−
4
f(6)=14
Answer:14
2
+
2(−3)
g(−3)=9−6
g(−3)=3
Answer:3
SAT Tip: Be careful with negatives! (−3)2=9
2
+
2(−3)
g(−3)=9−6
g(−3)=3
Answer:3
SAT Tip: Be careful with negatives! (−3)2=9
2a=12
a=6
Answer:a=6
Check:h(6)=2(6)+5=17 ✓
2a=12
a=6
Answer:a=6
Check:h(6)=2(6)+5=17 ✓
f(5)=15−7=8
Answer:f(5)=8
Key concept:f(5) means "plug in 5 for x." Function notation is just a way to name the output.
f(5)=15−7=8
Answer:f(5)=8
Key concept:f(5) means "plug in 5 for x." Function notation is just a way to name the output.
(x−1)(x−3)=0
Step 3: Apply zero product property:
x−1=0⟹x=1x−3=0⟹x=3
Check:f(1)=1−4+3=0 ✓ and f(3)=9−12+3=0 ✓
Answer:x=1 and x=3
SAT Context: These are the x-intercepts (zeros/roots) of the function's graph.
(x−1)(x−3)=0
Step 3: Apply zero product property:
x−1=0⟹x=1x−3=0⟹x=3
Check:f(1)=1−4+3=0 ✓ and f(3)=9−12+3=0 ✓
Answer:x=1 and x=3
SAT Context: These are the x-intercepts (zeros/roots) of the function's graph.
13
2a=12
a=6
Step 2: Find g(2a)=g(12):
g(12)=2(12)+1=25
Answer:g(2a)=25
Alternative approach: Notice g(2a)=2(2a)+1=4a+1. Since 2a=12, we get g(2a)=2(12)+1=25.
13
2a=12
a=6
Step 2: Find g(2a)=g(12):
g(12)=2(12)+1=25
Answer:g(2a)=25
Alternative approach: Notice g(2a)=2(2a)+1=4a+1. Since 2a=12, we get g(2a)=2(12)+1=25.
1=
4−
1=
3
Step 2: Now evaluate f(g(2))=f(3):
f(3)=2(3)+3=9
Answer:f(g(2))=9
Key concept: Composition of functions — work from the inside out. First evaluate g(2), then plug that result into f.
1=
4−
1=
3
Step 2: Now evaluate f(g(2))=f(3):
f(3)=2(3)+3=9
Answer:f(g(2))=9
Key concept: Composition of functions — work from the inside out. First evaluate g(2), then plug that result into f.