Practice Problems – Functions & Graphs – AP Calculus

Problem 01 Domain

Find the domain of \( f(x) = \frac{\sqrt{x-4}}{x} \).

Solution:
• Square root condition: \( x - 4 \ge 0 \) → \( x \ge 4 \)
• Denominator condition: \( x \neq 0 \)
• Intersection: \( x \ge 4 \) (automatically satisfies \( x \neq 0 \))
Domain: \( [4, \infty) \)

Problem 02 Domain

Find the domain of \( f(x) = \sqrt{x+2} \).

Solution:
• Square root condition: \( x + 2 \ge 0 \) → \( x \ge -2 \)
Domain: \( [-2, \infty) \)

Problem 03 Range

Find the range of \( f(x) = x^2 + 3 \).

Solution:
• \( x^2 \ge 0 \) for all real \( x \)
• \( x^2 + 3 \ge 3 \)
Range: \( [3, \infty) \)

Problem 04 Linear Function

Find the slope and y-intercept of \( f(x) = -4x + 7 \).

Solution:
• Slope \( m = -4 \)
• y-intercept \( b = 7 \)

Problem 05 Quadratic Function

Find the vertex of \( f(x) = x^2 + 4x - 5 \).

Solution:
• Complete the square: \( x^2 + 4x - 5 = (x+2)^2 - 9 \)
• Vertex: \( (-2, -9) \)

Problem 06 Rational Function

Find the domain of \( f(x) = \frac{3x}{x^2 - 9} \).

Solution:
• Denominator: \( x^2 - 9 = (x-3)(x+3) = 0 \) → \( x = 3 \) or \( x = -3 \)
Domain: \( (-\infty, -3) \cup (-3, 3) \cup (3, \infty) \)

Problem 07 Exponential Function

Evaluate \( f(3) \) if \( f(x) = 2 \cdot (1.5)^x \).

Solution:
• \( f(3) = 2 \cdot (1.5)^3 = 2 \cdot 3.375 = 6.75 \)

Problem 08 Logarithmic Function

Find the domain of \( f(x) = \ln(x-1) \).

Solution:
• Argument must be positive: \( x - 1 > 0 \) → \( x > 1 \)
Domain: \( (1, \infty) \)

Problem 09 Trigonometric Function

Find \( \sin\left(\frac{5\pi}{6}\right) \).

Solution:
• \( \frac{5\pi}{6} \) is in Quadrant II
• Reference angle: \( \frac{\pi}{6} \)
• Sine is positive in Quadrant II
• \( \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \)

Problem 10 Inverse Function

Find the inverse of \( f(x) = 3x - 5 \).

Solution:
• Let \( y = 3x - 5 \)
• Swap \( x \) and \( y \): \( x = 3y - 5 \)
• Solve for \( y \): \( 3y = x + 5 \) → \( y = \frac{x+5}{3} \)
• \( f^{-1}(x) = \frac{x+5}{3} \)

Problem 11 Piecewise Function

Given \( f(x) = \begin{cases} x^2, & x < 0 \\ 2x+1, & x \ge 0 \end{cases} \), find \( f(-2) \) and \( f(3) \).

Solution:
• \( f(-2) = (-2)^2 = 4 \)
• \( f(3) = 2(3) + 1 = 7 \)

Problem 12 Transformations

Describe the transformation from \( f(x) = x^2 \) to \( g(x) = (x-3)^2 + 2 \).

Solution:
• Inside \( (x-3) \) → shift right 3 units
• Outside \( +2 \) → shift up 2 units

Problem 13 Composition

If \( f(x) = x^2 \) and \( g(x) = 2x - 1 \), find \( (f \circ g)(x) \) and \( (g \circ f)(x) \).

Solution:
• \( (f \circ g)(x) = f(2x-1) = (2x-1)^2 = 4x^2 - 4x + 1 \)
• \( (g \circ f)(x) = g(x^2) = 2x^2 - 1 \)

Problem 14 Even Function

Determine if \( f(x) = x^4 - 2x^2 \) is even, odd, or neither.

Solution:
• \( f(-x) = (-x)^4 - 2(-x)^2 = x^4 - 2x^2 = f(x) \)
• Therefore, \( f \) is even

Problem 15 Odd Function

Determine if \( f(x) = x^5 - x \) is even, odd, or neither.

Solution:
• \( f(-x) = (-x)^5 - (-x) = -x^5 + x = -(x^5 - x) = -f(x) \)
• Therefore, \( f \) is odd

Problem 16 Neither Even nor Odd

Determine if \( f(x) = x^2 + x \) is even, odd, or neither.

Solution:
• \( f(-x) = x^2 - x \)
• Not equal to \( f(x) \) → not even
• Not equal to \( -f(x) = -x^2 - x \) → not odd
• Therefore, \( f \) is neither even nor odd

Problem 17 Trigonometric Identity

Simplify \( \sin^2 x + \cos^2 x \).

Solution:
• By the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \)

Problem 18 Trigonometric Transformations

Find the amplitude and period of \( y = 3\sin(2x) \).

Solution:
• Amplitude = \( |3| = 3 \)
• Period = \( \frac{2\pi}{|2|} = \pi \)

Problem 19 Absolute Value

Rewrite \( |x - 2| \) as a piecewise function.

Solution:
• \( |x - 2| = \begin{cases} x-2, & x \ge 2 \\ -(x-2), & x < 2 \end{cases} = \begin{cases} x-2, & x \ge 2 \\ -x+2, & x < 2 \end{cases} \)

Problem 20 Inverse Relationship

Simplify \( 10^{\log_{10} 5} \).

Solution:
• Since exponentials and logarithms are inverses: \( 10^{\log_{10} 5} = 5 \)

📝 Practice Problems