Question

the graph represents a functional relationship. which value is an input of the function? -14 -2 0 4

Explanation:

Step1: Recall function input definition

In a function, the input is the \( x \)-value (domain), and the output is the \( y \)-value (range). We need to check which of the given options is an \( x \)-value (input) that the graph passes through.

Step2: Analyze the graph's \( x \)-values

Looking at the graph, the \( x \)-axis values range (from the grid) and the line passes through \( x = 4 \) (wait, no, let's check the points. Wait, the graph has a line; let's see the options: -14, -2, 0, 4. Wait, the \( x \)-axis on the graph starts from, looking at the grid, the \( x \)-coordinates visible: the line goes through, for example, when \( x = 4 \), what's the \( y \)? Wait, no, let's check the options. Wait, the input is \( x \), so we need to see which of these \( x \)-values is in the domain (i.e., the graph has a point with that \( x \)). Wait, the graph's \( x \)-axis: the grid has \( x \) from, say, 0 to 18? Wait, no, the left side: the first \( x \)-grid line after 0 is 2, 4, 6, etc. Wait, the options: -14: is \( x = -14 \) on the graph? The graph's \( x \)-axis starts at, looking at the left, the \( x \)-values shown are from, maybe, negative? Wait, no, the \( x \)-axis is drawn with positive numbers (2,4,6,...18) and the left side (before 0) is not shown? Wait, no, the graph's \( x \)-axis: the origin is where? Wait, the \( y \)-axis is at \( x = 0 \)? Wait, no, the \( x \)-axis is horizontal, \( y \)-axis vertical. The first \( x \)-grid line to the right of \( y \)-axis is \( x = 2 \), then 4, 6, etc. Wait, the line passes through \( x = 4 \)? Wait, no, let's check the options again. Wait, the options are -14, -2, 0, 4. Wait, maybe I misread. Wait, the graph: the line starts at a point with \( x \) around, say, \( x = 0 \) (no, the first point is at \( x = 1 \) or \( x = 0 \)? Wait, the first point (the dot) is at \( x = 1 \) (maybe \( x = 1 \), \( y = 3 \))? Wait, no, the grid: each square is, say, 2 units? Wait, no, the \( y \)-axis has 8,6,4,2,0,-2,... so each grid line is 2 units? No, wait, the \( y \)-axis: from 8 to -14, with grid lines at 2-unit intervals? Wait, no, the \( x \)-axis: the labels are 2,4,6,8,10,12,14,16,18. So each grid square is 2 units in \( x \)? Wait, no, maybe each grid square is 2 units? Wait, no, the \( x \)-axis: the distance between \( x = 2 \) and \( x = 4 \) is 2 units, so each grid square is 2 units? Wait, no, maybe each grid square is 1 unit? Wait, the \( y \)-axis: 8,6,4,2,0,-2,... so each grid line is 2 units? No, that can't be. Wait, maybe each grid square is 2 units in \( y \) and 2 units in \( x \)? Wait, no, let's look at the line. The line passes through \( x = 4 \)? Wait, the options: 4 is an option. Wait, let's check the other options: -14: \( x = -14 \) is far left, not on the graph. -2: \( x = -2 \): is there a point with \( x = -2 \)? The graph's \( x \)-axis starts at, maybe, \( x = 0 \) (the \( y \)-axis) and goes right. So \( x = -2 \) and \( x = -14 \) are to the left of \( y \)-axis, which is not shown (the graph's \( x \)-axis labels start at 2,4,...). \( x = 0 \): is there a point at \( x = 0 \)? The line: when \( x = 0 \), what's \( y \)? The line passes through \( x = 4 \)? Wait, no, let's see the line's equation. Let's find two points. The dot is at (1, 3) (maybe \( x = 1 \), \( y = 3 \))? Wait, no, the grid: each square is 2 units? Wait, the \( y \)-axis: from 8 to -14, with grid lines at 2,4,6,... (positive) and -2,-4,... (negative). The \( x \)-axis: grid lines at 2,4,6,...18. So the line passes through, for example, (4, 0)? Wait, when \( x = 4 \), \( y = 0 \)? Wait, the line crosses the \( x \)-axis at \( x = 6 \)? Wait, no, let's check the options again. The options are -14, -2, 0, 4. Wait, maybe I made a mistake. Wait, the input is \( x \), so we need to see which \( x \) is in the domain. Wait, the graph's line: does it have a point with \( x = 4 \)? Let's check the options. -14: not on the graph (too far left). -2: not on the graph (left of \( y \)-axis, not shown). 0: is \( x = 0 \) on the graph? The line: when \( x = 0 \), what's \( y \)? The first point (the dot) is at \( x = 1 \) (maybe \( x = 1 \), \( y = 3 \)), then the line goes down to \( x = 4 \), \( y = 0 \)? Wait, no, the line passes through (4, 0)? Wait, the \( x \)-axis at \( x = 4 \) has \( y = 0 \)? Then \( x = 4 \) is an input (since it's an \( x \)-value the function takes). The other options: -14 (no), -2 (no), 0 (is there a point at \( x = 0 \)? The line at \( x = 0 \) would have \( y = 2 \)? Wait, no, let's calculate the slope. The dot is at (1, 3) (assuming \( x = 1 \), \( y = 3 \)) and then (4, 0)? Wait, slope is \( (0 - 3)/(4 - 1) = -1 \). So equation: \( y - 3 = -1(x - 1) \) → \( y = -x + 4 \). So when \( x = 4 \), \( y = 0 \); when \( x = 0 \), \( y = 4 \); when \( x = 6 \), \( y = -2 \); when \( x = 16 \), \( y = -12 \). Wait, so \( x = 4 \) is an input (since \( x = 4 \) gives \( y = 0 \), so \( x = 4 \) is in the domain). The other options: -14: \( y = -(-14) + 4 = 18 \), but the graph doesn't go that far left. -2: \( y = -(-2) + 4 = 6 \), but the graph's leftmost \( x \) is not -2 (the graph starts at \( x \approx 1 \) or so). 0: \( y = 4 \), but is \( x = 0 \) on the graph? The graph's \( x \)-axis: the \( y \)-axis is at \( x = 0 \), but the first grid line to the right is \( x = 2 \). Wait, maybe the grid is 2 units per square? So \( x = 0 \) is the \( y \)-axis, \( x = 2 \) is the first square, \( x = 4 \) the second, etc. So the line passes through \( x = 4 \) (which is a grid line), so \( x = 4 \) is an input.

Answer:

4 (the option is "4", so the answer is 4, corresponding to the option "4".)