College Algebra Study Guide: Functions, Domains, Ranges, and Transformations

Q1. Determine whether the relation is a function. Give the domain and the range for the relation: {(2,3), (4,5), (8,8)}

Background

Topic: Functions and Relations

This question tests your understanding of what makes a relation a function, and how to identify the domain and range from a set of ordered pairs.

Key Terms:

• Function: A relation in which each input (x-value) corresponds to exactly one output (y-value).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

Step-by-Step Guidance

1. List all the x-values and y-values from the given pairs.

2. Check if any x-value is repeated with a different y-value. If not, the relation is a function.

3. Identify the domain by collecting all unique x-values.

4. Identify the range by collecting all unique y-values.

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Q2. Use the graph of f to determine each of the following: domain, range, zeros, intervals of increase/decrease, relative maximum/minimum, and values for which f(x) is positive or negative.

Background

Topic: Graph Analysis of Functions

This question tests your ability to interpret a function's graph and extract information such as domain, range, zeros, and intervals of increase/decrease.

Key Terms:

• Domain/Range: Interval notation is often used for continuous graphs.

• Zeros: x-values where f(x) = 0.

• Increasing/Decreasing: Intervals where the graph goes up or down.

• Relative Maximum/Minimum: Highest/lowest points in a local region.

Step-by-Step Guidance

1. Examine the graph to determine the domain (x-values covered) and range (y-values covered).

2. Identify the zeros by finding where the graph crosses the x-axis.

3. Look for intervals where the graph is rising (increasing) or falling (decreasing).

4. Find any relative maximum or minimum points by checking peaks and valleys.

5. Determine where f(x) is positive (above x-axis) or negative (below x-axis).

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Q3. Find the difference quotient of f, that is, find \frac{f(x+h)-f(x)}{h}, h \neq 0, for the function f(x) = x^2 - 5x + 5.

Background

Topic: Difference Quotient

This question tests your ability to compute the difference quotient, which is foundational for understanding rates of change and calculus concepts.

Key Formula:

• $\frac{f(x+h)-f(x)}{h}$

Step-by-Step Guidance

1. Calculate $f(x+h)$ by substituting $x+h$ into the function.

2. Subtract $f(x)$ from $f(x+h)$.

3. Divide the result by $h$ and simplify the expression.

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Q4. The functions f and g are graphed in the same rectangular coordinate system. If g is obtained from f through a sequence of transformations, find an equation for g.

Background

Topic: Function Transformations

This question tests your understanding of how to write the equation of a function after it has been transformed (shifted, reflected, stretched, etc.).

Key Terms:

• Transformation: Operations such as translation, reflection, and scaling applied to functions.

Step-by-Step Guidance

1. Identify the original function f(x) from the graph.

2. Observe how the graph of g(x) differs from f(x) (e.g., shifts left/right, up/down).

3. Write the equation for g(x) based on the observed transformation(s).

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Q5. Find the domain of the function f(x) = x^2 - 13x + 4.

Background

Topic: Domain of Polynomial Functions

This question tests your ability to determine the domain of a polynomial function, which is typically all real numbers.

Key Terms:

• Domain: The set of all possible input values (x-values) for which the function is defined.

Step-by-Step Guidance

1. Recognize that polynomials are defined for all real numbers.

2. Express the domain in interval notation.

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Q6. Find the domain of the function f(x) = \sqrt{x+5}.

Background

Topic: Domain of Radical Functions

This question tests your ability to find the domain of a function involving a square root, which requires the radicand to be non-negative.

Key Terms:

• Radicand: The expression inside the square root.

• Domain: Values of x for which the radicand is greater than or equal to zero.

Step-by-Step Guidance

1. Set the radicand $x+5 \geq 0$ and solve for x.

2. Express the solution in interval notation.

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Q7. Find the domain of the function f(x) = \frac{1}{\sqrt{x+11}}.

Background

Topic: Domain of Rational and Radical Functions

This question tests your ability to find the domain of a function with both a square root and a denominator, requiring the radicand to be positive.

Key Terms:

• Denominator: Cannot be zero.

• Radicand: Must be positive for the square root in the denominator.

Step-by-Step Guidance

1. Set $x+11 > 0$ to ensure the denominator is defined and positive.

2. Solve for x and write the domain in interval notation.

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Q8. Find the domain of the function g(x) = \frac{\sqrt{x-2}}{x-9}.

Background

Topic: Domain of Rational Functions with Radicals

This question tests your ability to find the domain of a function with a square root in the numerator and a variable in the denominator.

Key Terms:

• Numerator: The radicand must be non-negative.

• Denominator: Cannot be zero.

Step-by-Step Guidance

1. Set $x-2 \geq 0$ to ensure the square root is defined.

2. Set $x-9 \neq 0$ to ensure the denominator is not zero.

3. Combine the restrictions to write the domain in interval notation.

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Q9. Find the domain of the function f(x) = \frac{3x+10}{x^3+2x^2-9x-18}.

Background

Topic: Domain of Rational Functions

This question tests your ability to find the domain of a rational function by identifying values that make the denominator zero.

Key Terms:

• Rational Function: A function that is the ratio of two polynomials.

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

Step-by-Step Guidance

1. Set the denominator equal to zero and solve for x.

2. Exclude these values from the domain.

3. Express the domain in interval notation, omitting the excluded values.

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