Precalc. AB Semester Final Part 1

Front

What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

Front

How do you find the horizontal asymptote of a rational function?

Back

For a rational function of the form \( f(x) = \frac{a_n x^n + ...}{b_m x^m + ...} \), if n < m, the horizontal asymptote is y = 0; if n = m, the horizontal asymptote is y = \( \frac{a_n}{b_m} \); if n > m, there is no horizontal asymptote.

Front

What is synthetic division?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form \( x - c \).

Front

What is the logarithm of a number?

Back

The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, if \( b^y = x \), then \( \log_b(x) = y \).

Front

How do you evaluate \( \log_3(\sqrt[5]{\frac{1}{9}}) \)?

Back

First, rewrite \( \sqrt[5]{\frac{1}{9}} \) as \( \left(\frac{1}{9}\right)^{1/5} = \left(3^{-2}\right)^{1/5} = 3^{-2/5} \). Thus, \( \log_3(3^{-2/5}) = -\frac{2}{5} \).

Front

What is a slant (oblique) asymptote?

Back

A slant asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator in a rational function.

Front

How do you find the slant asymptote of a rational function?

Back

To find the slant asymptote, perform polynomial long division of the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the slant asymptote.

Front

What is function composition?

Back

Function composition is the process of applying one function to the results of another function. If \( f(x) \) and \( g(x) \) are functions, then the composition is denoted as \( g(f(x)) \).

Front

How do you find \( g(f(x)) \) for \( f(x) = 4x^2 + 3x + 5 \) and \( g(x) = 3x - 8 \)?

Back

Substitute \( f(x) \) into \( g(x) \): \( g(f(x)) = 3(4x^2 + 3x + 5) - 8 = 12x^2 + 9x + 15 - 8 = 12x^2 + 9x + 7 \).

Front

What is the definition of a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.