Precalculus Study Guide: Functions, Graphs, and Transformations

Q1. Determine whether the equation defines y as a function of x.

Background

Topic: Definition of a Function

This question tests your understanding of what it means for an equation to define y as a function of x. A function assigns exactly one output (y-value) for each input (x-value).

Key Terms:

• Function: A relation where each input has exactly one output.

• Vertical Line Test: A graphical method to determine if a relation is a function.

Step-by-Step Guidance

1. For each equation, try to solve for y in terms of x. If you get more than one value of y for a single x, it is not a function.

2. For equations like $y = x^2 - 2x$, check if every x gives only one y.

3. For equations like $y^2 = 3x - 4$, try to solve for y and see if you get two possible values for some x.

4. For equations with x and y on the same side, consider if the graph would pass the vertical line test.

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Q2. For $f(x) = -x^2 + 2x - 3$, find (a) $f(0)$, (b) $f(-1)$, (c) $f(3)$

Background

Topic: Evaluating Functions

This question tests your ability to substitute values into a function and simplify to find the output.

Key Formula:

$f(x) = -x^2 + 2x - 3$

Step-by-Step Guidance

1. For each part, substitute the given value of x into the function.

2. Calculate $-x^2$ for the given x.

3. Add $2x$ to the result.

4. Subtract 3 from the sum to get $f(x)$.

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Q3. Find the domain of each function.

Background

Topic: Domain of Functions

This question tests your understanding of the domain, which is the set of all real x-values for which the function is defined.

Key Concepts:

• For rational functions, exclude x-values that make the denominator zero.

• For even roots (like square roots), the expression under the root must be non-negative.

Step-by-Step Guidance

1. For each function, identify any restrictions (denominator cannot be zero, even root must be non-negative).

2. Solve the equation or inequality to find excluded values or intervals.

3. Express the domain in interval notation.

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Q4. For $f(x) = 2x - 3$ and $g(x) = 2x^2$, find:

• (a) $(f + g)(x)$

• (b) $(f - g)(x)$

• (c) $(f \\cdot g)(2)$

• (d) $\\left( \\frac{f}{g} \\right)(3)$

Background

Topic: Operations with Functions

This question tests your ability to perform addition, subtraction, multiplication, and division with functions, as well as evaluate the result at a specific value.

Key Formulas:

• $(f + g)(x) = f(x) + g(x)$

• $(f - g)(x) = f(x) - g(x)$

• $(f \\cdot g)(x) = f(x) \\cdot g(x)$

• $\\left( \\frac{f}{g} \\right)(x) = \\frac{f(x)}{g(x)}$ (make sure $g(x) \\neq 0$)

Step-by-Step Guidance

1. For each operation, write out the expressions for $f(x)$ and $g(x)$.

2. For (a) and (b), combine the expressions as indicated (add or subtract).

3. For (c), multiply $f(2)$ and $g(2)$.

4. For (d), divide $f(3)$ by $g(3)$, ensuring $g(3) \\neq 0$.

Try solving on your own before revealing the answer!