College Algebra: Practice Problems on Functions, Asymptotes, and Logarithms

Equations and Asymptotes of Functions

Identifying Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are important in understanding the behavior of rational and certain transcendental functions.

• Vertical Asymptotes: Occur where the function is undefined due to division by zero.

• Horizontal Asymptotes: Indicate the value the function approaches as x tends to infinity or negative infinity.

• Oblique (Slant) Asymptotes: Occur when the degree of the numerator is exactly one more than the denominator in a rational function.

Example: For $f(x) = \frac{1}{x-2}$, the vertical asymptote is $x=2$ and the horizontal asymptote is $y=0$.

Rational Functions

Analyzing Rational Functions

Rational functions are quotients of polynomials. Their properties can be determined by analyzing their numerators and denominators.

• Domain: All real numbers except where the denominator is zero.

• x-intercept: Set the numerator equal to zero and solve for x.

• y-intercept: Evaluate the function at $x=0$.

• Vertical Asymptotes: Set the denominator equal to zero and solve for x.

• Horizontal Asymptotes: Compare the degrees of the numerator and denominator:

  • If degree numerator < degree denominator: $y=0$

  • If degrees are equal: $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$

  • If degree numerator > degree denominator: No horizontal asymptote (may have slant asymptote)

Example: For $f(x) = \frac{3x-2}{x+1}$:

• Domain: $x \neq -1$

• x-intercept: $3x-2=0 \implies x=\frac{2}{3}$

• y-intercept: $f(0) = \frac{-2}{1} = -2$

• Vertical asymptote: $x=-1$

• Horizontal asymptote: $y=3$ (degrees are equal, leading coefficients are 3 and 1)

Inverse Functions

Definition and Properties

An inverse function reverses the effect of the original function. If $f(x)$ is one-to-one, then its inverse $f^{-1}(x)$ exists and satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• To find the inverse, solve $y = f(x)$ for $x$ and interchange $x$ and $y$.

• The graph of $f^{-1}(x)$ is the reflection of $f(x)$ across the line $y=x$.

Example: If $f(x) = 2x+3$, then $f^{-1}(x) = \frac{x-3}{2}$.

Logarithmic and Exponential Functions

Properties of Logarithms

Logarithms are the inverses of exponential functions. They have several important properties:

• $\log_b(MN) = \log_b M + \log_b N$

• $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

• $\log_b(M^k) = k\log_b M$

• $\log_b b = 1$ and $\log_b 1 = 0$

Example: $\log_2 8 = 3$ because $2^3 = 8$.

Solving Logarithmic and Exponential Equations

• To solve $a^x = b$, take the logarithm of both sides: $x = \log_a b$.

• To solve $\log_b x = y$, rewrite as $x = b^y$.

Example: Solve $2^x = 16$. $x = \log_2 16 = 4$.

Graphing and One-to-One Functions

One-to-One Functions

A function is one-to-one if each output is produced by exactly one input. This property is necessary for a function to have an inverse.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

Example: $f(x) = x^2$ is not one-to-one, but $f(x) = x^3$ is one-to-one.

Graphing Rational and Logarithmic Functions

• Identify intercepts and asymptotes before sketching the graph.

• Plot key points and use symmetry if applicable.

• For logarithmic functions, the vertical asymptote is at $x=0$ for $f(x) = \log_b x$.

Combining and Expanding Logarithms

Expressing Sums and Differences as Single Logarithms

• Use properties of logarithms to combine or expand expressions.

Example: $\log_2 7 + \log_2 5 = \log_2(35)$

Example: $\ln\left(\frac{x}{y}\right) = \ln x - \ln y$

Solving Logarithmic and Exponential Equations

Exact Solutions

• Isolate the logarithmic or exponential part of the equation.

• Apply the appropriate inverse operation (logarithm or exponentiation).

• Check for extraneous solutions, especially when dealing with logarithms.

Example: Solve $\log_3(x-2) = 4$. $x-2 = 3^4 = 81 \implies x = 83$.

Summary Table: Properties of Logarithms

Property	Formula	Example
Product Rule	$\log_b(MN) = \log_b M + \log_b N$	$\log_2(8 \times 4) = \log_2 8 + \log_2 4$
Quotient Rule	$\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$	$\log_3\left(\frac{9}{3}\right) = \log_3 9 - \log_3 3$
Power Rule	$\log_b(M^k) = k\log_b M$	$\log_5(25) = 2\log_5 5$

Additional info: These study notes are based on a set of practice problems covering rational functions, asymptotes, inverse functions, logarithms, and graphing, which are all core topics in College Algebra (Chapters 2, 3, and 4).