Q1. Evaluate the piecewise function at and .

Background

Topic: Piecewise Functions

This question tests your ability to evaluate a function defined by different expressions depending on the value of . Piecewise functions are important in calculus for modeling situations where a rule changes based on input.

Key Terms and Formulas:

• Piecewise function: A function defined by multiple sub-functions, each with its own domain.

• To evaluate: Substitute the given value into the correct expression based on the domain.

Step-by-Step Guidance

1. Identify which part of the piecewise function applies for each value of :

   • For , check if or .

   • For , check if or .

2. For each value, substitute it into the appropriate expression:

   • If , use .

   • If , use .

3. Set up the calculations for each case:

   • For :

   • For :

Try solving on your own before revealing the answer!

Q2. Evaluate the piecewise function at and .

Background

Topic: Piecewise Functions

This question tests your understanding of how to evaluate a function with different rules for different intervals of .

Key Terms and Formulas:

• Piecewise function: Multiple expressions, each valid for a specific range of .

• To evaluate: Substitute the value of into the correct expression.

Step-by-Step Guidance

1. Determine which expression to use for each value:

   • For , check if or .

   • For , check if or .

2. Substitute each value into the appropriate expression:

   • If , use .

   • If , use .

3. Set up the calculations:

   • For :

   • For :

Try solving on your own before revealing the answer!

Q3. Graph each piecewise function, then identify the domain and range.

Background

Topic: Graphing Piecewise Functions

This question tests your ability to graph functions defined by different expressions over different intervals, and to determine the domain (all possible values) and range (all possible values).

Key Terms and Formulas:

• Domain: The set of all values for which the function is defined.

• Range: The set of all values the function can take.

• Piecewise function: Graph each segment according to its rule and interval.

Step-by-Step Guidance

1. For each function, identify the intervals and the corresponding expressions.

2. For each interval, plot the graph using the correct expression. Pay attention to open or closed endpoints based on the inequalities.

3. Combine the segments to form the complete graph.

4. Analyze the graph to determine the domain and range.

Try solving on your own before revealing the answer!