A function is defined as a relation where each input corresponds to exactly one output. This property distinguishes functions from general relations that might associate an input with more than one output.

The square root function is only defined for non-negative values since the square root of a negative number is not a real number. Therefore, the domain of √x is x ≥ 0.

To evaluate the function f(x) = x + 3 at x = 2, substitute 2 into the function to get 2 + 3, which equals 5. This demonstrates basic function evaluation.

A quadratic function is typically defined as a second-degree polynomial in the form ax² + bx + c with a ≠ 0. The function f(x) = x² + 2x + 1 fits this definition, making it quadratic.

A linear function is generally expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. This form produces a straight line when graphed.

The function 1/(x - 4) is undefined when the denominator equals zero. Setting x - 4 = 0 yields x = 4, which must be excluded from the domain.

Since 16 can be expressed as 2^4, the equation 2^x = 16 implies x must equal 4. This relies on understanding the properties of exponents.

To find (f ∘ g)(2), first calculate g(2) which is 2² = 4. Then evaluate f(4): 2 × 4 - 3 = 5.

The vertex of a parabola defined by y = ax² + bx + c is found using the formula x = -b/(2a). For a = -1 and b = 4, x = 2, and substituting back gives y = 1. Thus, the vertex is (2, 1).

Replacing x with x - 2 in the function shifts the graph horizontally to the right by 2 units. Adding 3 shifts it vertically upward by 3 units, combining to form the given transformation.

The equation log₂(x) = 3 means that 2³ = x. Since 2³ equals 8, x must be 8. This uses the basic definition of logarithms.

One of the fundamental properties of logarithms is that the log of a product is the sum of the logs. Therefore, log(a * b) simplifies to log(a) + log(b).

To find the inverse of f(x) = 3x - 5, swap x and y to obtain x = 3y - 5 and then solve for y. The solution y = (x + 5)/3 gives the inverse function.

The amplitude of a sine function is the absolute value of the coefficient in front of sin(x). In y = 5 sin(x), the coefficient is 5, so the amplitude is 5.

The composition (f ∘ g)(x) means f(g(x)). Given g(x) = √x, applying f gives (√x)², which simplifies to x as long as x is non-negative. This illustrates function composition and domain consideration.

For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients (2/1 = 2). Vertical asymptotes occur where the denominator equals zero, which happens at x = 1 and x = -1.

Factoring the quadratic gives (x - 2)(x - 3) < 0. The product of two factors is negative when one is positive and the other is negative, which occurs when x is between 2 and 3.

The period of a cosine function in the form cos(Bx) is given by 2π/B. Here, B is 4, so the period is 2π/4 = π/2.

The series increases by a common difference of 3. Calculating the number of terms using n = ((last term - first term)/difference) + 1 gives 10 terms, and applying the sum formula S = n/2 × (first term + last term) results in 155.

Using the chain rule to differentiate f(x) = 3e^(2x), the derivative of e^(2x) is 2e^(2x). Multiplying by the constant 3 gives f'(x) = 6e^(2x). This demonstrates the application of the chain rule.