Question

part ii questions: answer all questions in this part. clearly indicate the necessary steps and explain your reasoning. for all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. a function f(x) is shown graphed. answer the following (a) what is the value of f(5)? justify. 3 points (b) state the domain and range of the function. 3 points domain: range: (c) give a domain interval over which the function is only increasing 3 points

Explanation:

(a)

Step1: Locate x=5 on graph

Find the point where \( x = 5 \) on the function's graph.

Step2: Determine y - value

At \( x = 5 \), the corresponding \( y \) - value (which is \( f(5) \)) from the graph is 9. Justification: On the graph of a function \( y = f(x) \), the value of \( f(a) \) is the \( y \) - coordinate of the point where \( x=a \). For \( x = 5 \), we look at the graph and see that the point has a \( y \) - coordinate of 9.

Step1: Find domain

The domain of a function is the set of all \( x \) - values for which the function is defined. Looking at the graph, the left - most point has an \( x \) - value of \( - 8 \) and the right - most point has an \( x \) - value of \( 9 \). So the domain is all real numbers from \( - 8 \) to \( 9 \), or in interval notation \( [-8,9] \).

Step2: Find range

The range of a function is the set of all \( y \) - values that the function takes. Looking at the graph, the lowest \( y \) - value (minimum) is \( - 4 \) and the highest \( y \) - value (maximum) is \( 9 \). So the range is all real numbers from \( - 4 \) to \( 9 \), or in interval notation \( [-4,9] \).

Step1: Analyze increasing intervals

A function is increasing when as \( x \) increases, \( y \) also increases. Looking at the graph, we can see that from \( x=-3 \) to \( x = 5 \) (or any sub - interval within this where the function is rising), the function is increasing. For example, the interval \( [-3,5] \): as \( x \) moves from \( - 3 \) to \( 5 \) along the \( x \) - axis, the \( y \) - values of the function are increasing.

Answer:

\( f(5)=9 \)

(b)