Question

1. graph what is the domain of the function? options: ( x leq 2 ), ( x geq 2 ), ( -4 < x leq -2 ), ( -4 leq x leq -2 )
2. karol is constructing a table of values that satisfies the definition of a function.

input: -24, 31, 1, -5, 22, -2, 28
output: -26, -22, -10, -3, -2, 6, 6, 24 (empty cell)
which number(s) can be placed in the empty cell so that the table of values satisfies the definition of function? circle all that apply.
options: a. -6, b. -2, c. 1, d. 3, e. 22, f. 28

1. graph what is ( f(4) )? options: a. 12, b. -1, c. 2, d. 1, e. 8, f. none of these

Explanation:

Question 39

Step1: Recall domain definition

The domain of a function is the set of all possible \( x \)-values (input values) for which the function is defined. From the graph, we observe the leftmost point (starting point) and the direction. The graph has a line segment with the rightmost \( x \)-value (the endpoint) at \( x = 2 \) (or looking at the grid, the endpoint is at \( x \leq 2 \) as the line extends to the left from \( x = 2 \)). Wait, actually, looking at the graph, the line starts from the left (higher \( x \)?) Wait, no, the graph: the left end is at some \( x \), but the right end (the solid dot) is at \( x = 2 \)? Wait, no, the grid: the \( x \)-axis, the solid dot is at \( x = 2 \)? Wait, the options: \( x \leq 2 \), \( x \geq 2 \), \( -4 < x \leq -2 \), \( -4 \leq x \leq -2 \). Wait, maybe I misread. Wait, the graph: the line is from the left (maybe \( x \leq 2 \))? Wait, the correct domain: the function's graph is a line segment where the \( x \)-values go from the left (infinite? No, the graph has a starting point? Wait, the graph shows a line with a solid dot at the right end (lower \( x \)?) Wait, no, the grid: the \( x \)-axis, the solid dot is at \( x = 2 \)? Wait, the options: the first option is \( x \leq 2 \). Let's check the graph: the line is going from the left (higher \( x \)) to the right (lower \( x \))? No, the \( y \)-axis is vertical. Wait, the graph: the left end is at \( x = -4 \)? No, the options have \( -4 \) related. Wait, maybe the graph is a line segment from \( x = -4 \) to \( x = -2 \)? No, the options: the first option is \( x \leq 2 \). Wait, maybe the graph is a line with domain \( x \leq 2 \). So the correct answer is \( x \leq 2 \).

Step1: Recall function definition

A function is a relation where each input (x - value) has exactly one output (y - value). So, the empty cell is an input, so we need to choose an input that is not already in the "Input" column.

Step2: List existing inputs

Existing inputs: -24, 31, 1, -5, 22, -2, 28.

Step3: Check each option

• Option A: -6. Not in existing inputs. Valid.
• Option B: -2. Already in inputs (input -2 has output 6). Invalid (duplicate input, would have two outputs? No, input -2 already has output 6, so adding -2 as input again would violate function definition (same input, different output? Wait, no, the output for the new cell is 24. Wait, no: the input is the empty cell, output is 24. So the input (x) must be unique. So existing inputs: -24, 31, 1, -5, 22, -2, 28. So input must not be any of these.

• Option B: -2. Already in inputs. So if we put -2 as input, then input -2 would have two outputs: 6 and 24. Which violates function definition. So B is invalid.

• Option C: 1. Already in inputs (input 1 has output -10). So invalid.

• Option D: 3. Not in existing inputs. Valid.

• Option E: 22. Already in inputs (input 22 has output -2). Invalid.

• Option F: 28. Already in inputs (input 28 has output 6). Invalid.

So valid options are A (-6) and D (3).

Step1: Recall function notation

\( f(4) \) means the value of the function when \( x = 4 \). We look at the graph of the function.

Step2: Find \( x = 4 \) on the graph

On the graph, when \( x = 4 \), we find the corresponding \( y \)-value (output). From the graph, at \( x = 4 \), the point is at \( y = 1 \)? Wait, no, looking at the graph: the line passes through (4,1)? Wait, the graph has a point at (4,1)? Wait, the options: a.12, b.-1, c.2, d.1, e.8, f.none. Let's check the graph: the line goes through (4,1)? Wait, the grid: x=4, y=1? Yes, the point at x=4 is at y=1. So \( f(4) = 1 \).

Answer:

\( x \leq 2 \) (the first option)

Question 40