Question

1. which of the following graphs is a function of x? select all that apply

a.

b.

c.

d.

e.

f.

Explanation:

To determine if a graph represents a function of \( x \), we use the Vertical Line Test: a graph is a function if no vertical line intersects the graph more than once.

Step 1: Analyze Graph a

Graph a is a spiral. A vertical line (e.g., \( x = 1 \)) will intersect the spiral multiple times. So, it is not a function.

Step 2: Analyze Graph b

Graph b is a curve. Any vertical line will intersect this curve at most once. So, it passes the Vertical Line Test and is a function.

Step 3: Analyze Graph c

Graph c is a straight line (linear). A vertical line will intersect this line at most once. So, it passes the Vertical Line Test and is a function.

Step 4: Analyze Graph d

Graph d has a "loop" or curve that a vertical line will intersect more than once (e.g., a vertical line through the middle of the loop). So, it fails the Vertical Line Test and is not a function.

Step 5: Analyze Graph e

Graph e is a curve (exponential-like). Any vertical line will intersect this curve at most once. So, it passes the Vertical Line Test and is a function.

Step 6: Analyze Graph f

Graph f is a vertical line ( \( x = \) some constant). A vertical line (itself) intersects infinitely, and other vertical lines won't intersect, but by definition, a vertical line does not represent a function (fails Vertical Line Test as it intersects itself infinitely). So, it is not a function.

Answer:

b. The graph, c. The graph, e. The graph