Bluebook SAT Test Algebra and Functions 2 — Questions and Answers
Question 1: If f(x) = 4 - x² and g(x) = 2x, what is (f + g)(3)?
- 1 (Correct answer)
- 7
- -5
- 11
Correct answer: 1
f(3) = 4-9 = -5. g(3) = 6. (f+g)(3) = -5+6 = 1.
(f+g)(x) = f(x)+g(x). At x=3: f(3) = 4-(3²) = 4-9 = -5. g(3) = 2(3) = 6. Therefore (f+g)(3) = f(3)+g(3) = -5+6 = 1. Function arithmetic follows the same rules as regular arithmetic — just evaluate each function at the given input, then combine.
Question 2: The linear function f satisfies f(0) = 5 and f(3) = -1. What is f(9)?
- -13 (Correct answer)
- -11
- 3
- -7
Correct answer: -13
Slope = (-1-5)/(3-0) = -2. f(x) = -2x+5. f(9) = -18+5 = -13.
A linear function has the form f(x) = mx+b. Using the two points: slope m = (f(3)-f(0))/(3-0) = (-1-5)/3 = -6/3 = -2. Since f(0) = 5, the y-intercept b = 5. So f(x) = -2x+5. At x=9: f(9) = -2(9)+5 = -18+5 = -13. Check: f(0)=5 ✓, f(3)=-6+5=-1 ✓.
Question 3: Which function has a range of all real numbers?
- f(x) = x³ (Correct answer)
- f(x) = x²
- f(x) = √x
- f(x) = 1/x
Correct answer: f(x) = x³
x³ can produce any real number as output (range = all reals). x² only produces non-negatives, √x only non-negatives, and 1/x cannot produce 0.
f(x) = x³: As x → -∞, x³ → -∞; as x → +∞, x³ → +∞. Every real number y can be achieved with x = y^(1/3). Range: all reals. f(x) = x²: Range is y ≥ 0 (squares are non-negative). f(x) = √x: Defined only for x ≥ 0, range is y ≥ 0. f(x) = 1/x: Never equals zero, range is all reals except 0. Only f(x) = x³ has range equal to all real numbers.
Question 4: If f(2x + 1) = 4x² + 4x - 3, what is f(5)?
- 21 (Correct answer)
- 17
- 13
- 9
Correct answer: 21
Set 2x+1=5: x=2. f(5) = 4(4)+4(2)-3 = 16+8-3 = 21.
We need f(5), so we set the argument 2x+1 = 5, giving x = 2. Substituting x=2 into the right side: f(5) = 4(2²)+4(2)-3 = 4(4)+8-3 = 16+8-3 = 21. This technique — matching the input to find the corresponding x — is essential for these 'function of a function' SAT problems.
Question 5: The graph of y = f(x) is shifted 3 units left and 2 units down. Which equation represents the new graph?
- y = f(x + 3) - 2 (Correct answer)
- y = f(x - 3) + 2
- y = f(x + 3) + 2
- y = f(x - 3) - 2
Correct answer: y = f(x + 3) - 2
Shifting left by 3 replaces x with (x+3). Shifting down by 2 subtracts 2. Result: y = f(x+3) - 2.
Function transformations: A horizontal shift of h units left replaces x with (x+h) in the function. A vertical shift of k units down subtracts k from the output. Starting with y=f(x): shift left 3 → y=f(x+3). Then shift down 2 → y=f(x+3)-2. Common error: students reverse the sign for horizontal shifts. Moving left increases the input value, hence x+3 (not x-3).
Question 6: For f(x) = (x + 1)/(x - 1), which value is NOT in the domain of f?
- 1 (Correct answer)
- -1
- 0
- 2
Correct answer: 1
The denominator x-1 = 0 when x = 1. So x = 1 is excluded from the domain.
The domain of a rational function excludes values where the denominator equals zero. For f(x) = (x+1)/(x-1): denominator = x-1. Setting x-1 = 0 gives x = 1. The domain is all real numbers except x = 1 (written as x ∈ (-∞,1)∪(1,∞)). Note: x=-1 is in the domain (numerator becomes 0, making f(-1)=0/(-2)=0, which is valid).
Question 7: If f(x) = 2^x and g(x) = log₂(x), what is f(g(16))?
- 16 (Correct answer)
- 4
- 256
- 8
Correct answer: 16
g(16) = log₂(16) = 4. f(4) = 2⁴ = 16. f and g are inverses, so f(g(16)) = 16.
g(x) = log₂(x) and f(x) = 2^x are inverse functions because f(g(x)) = 2^(log₂x) = x. So f(g(16)) = 16 directly. To verify step by step: g(16) = log₂(16) = log₂(2⁴) = 4. Then f(4) = 2⁴ = 16. This illustrates the fundamental inverse relationship between exponential and logarithmic functions.
Question 8: The graph of f(x) passes through (0, 3) and (2, 11). If f is linear, what is f(5)?
- 23 (Correct answer)
- 19
- 27
- 15
Correct answer: 23
Slope = (11-3)/(2-0) = 4. f(x) = 4x+3. f(5) = 20+3 = 23.
With points (0,3) and (2,11): slope m = (11-3)/(2-0) = 8/2 = 4. Since (0,3) gives y-intercept b=3, the equation is f(x) = 4x+3. Evaluating at x=5: f(5) = 4(5)+3 = 20+3 = 23. The linear function increases by 4 for each unit increase in x.
Bluebook SAT Test Algebra and Functions 2
If f(x) = 4 - x² and g(x) = 2x, what is (f + g)(3)?