Understanding Functions and Their Graphs

What is the definition of a function?

• Has inputs and outputs

• Every input has only ONE output

• Inputs have different outputs every time

• x-values and y-values

Answer explanation

A function is defined such that every input (x-value) corresponds to exactly one output (y-value). This means that for each input, there cannot be multiple outputs, making 'Every input has only ONE output' the correct definition.

• -3

• -1

• 1

• 3

Answer explanation

To find f(-3), locate -3 on the x-axis of the graph. The corresponding y-value is -3, which means f(-3) = -3. Therefore, the correct answer is -3.

• 2

• 4

• 8

Answer explanation

The y-intercept of a function is found by evaluating it at x=0. For f(x)=2(4)^x, f(0)=2(4)^0=2(1)=2. Thus, the y-intercept is 2.

Write the recursive rule for this situation.

Answer explanation

The correct recursive rule is f(n) = f(n - 1) + 7, n ≥ 2, indicating that each term is 7 more than the previous term. The other options either add 58 or use incorrect increments.

What is the common difference?

• −16

• 84

• 16

• 0

Answer explanation

The common difference in an arithmetic sequence is found by subtracting any term from the subsequent term. If the difference is consistently 16, then the correct answer is 16.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities. Which equation matches the graph?

Answer explanation

The graph of y = 2^{-x} shows exponential decay, starting high and approaching zero as x increases. This matches the behavior of the correct choice, indicating a decrease in quantity as x increases.

What are the equations of the asymptotes of the given graph?

Answer explanation

The correct asymptotes are found by analyzing the graph's behavior. The vertical asymptotes are at x=2 and x=-2, while the horizontal asymptote is at y=1, making the correct choice x=2, x=-2, and y=1.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. What is the domain of the graph?

• All real numbers

• None of these

Answer explanation

The domain of a function represents all possible input values (x-values). If the graph shows no restrictions on x, then the domain is all real numbers, making 'All real numbers' the correct choice.