Answered


$x$		$f(x)$

-2		20

-1		0

0		-6

1		-4

2		0

3		0






Which is an $x$-intercept of the continuous function in the table?

A(-1, 0)
B(0, -6)
C(-6, 0)
D(0, -1)

Verified Answer :

  • Understand that an x-intercept is a point where the function value is zero, denoted as \((x, 0)\).
  • Examine the provided table to find all occurrences where \(f(x) = 0\), which are \((-1, 0)\), \((2, 0)\), and \((3, 0)\).
  • Compare these points to the given options to find the match.
  • Conclude that \((-1, 0)\) is the correct x-intercept. Final answer: \(\boxed{(-1, 0)}\).

Explanation

  1. Understanding X-Intercepts

An x-intercept of a function \(f(x)\) is a point where the graph crosses the x-axis. Mathematically, this occurs at any point \((x, 0)\) such that \(f(x) = 0\). Our goal is to identify which of the provided options represents one of these points based on the given table of values.

  1. Identifying X-Intercepts from the Table

Let's examine the provided table to identify all points where \(f(x) = 0\):

\[ @@TABLE_TOKEN_0@@ \]

By scanning the table, we find that \(f(x) = 0\) at the following values of \(x\):

  • At \(x = -1\), \(f(-1) = 0\), which gives the point \((-1, 0)\).
  • At \(x = 2\), \(f(2) = 0\), which gives the point \((2, 0)\).
  • At \(x = 3\), \(f(3) = 0\), which gives the point \((3, 0)\).

These are the x-intercepts of the function.

  1. Comparing with Given Options

Now we compare our identified x-intercepts with the given options:

  • \((-1, 0)\): This matches one of the points we found in our table analysis.
  • \((0, -6)\): This represents a point where \(x=0\), making it a y-intercept.
  • \((-6, 0)\): This value of \(x\) is not provided in our table as having a value of \(f(x) = 0\).
  • \((0, -1)\): This point is not an intercept because \(f(0) = -6\).

Therefore, \((-1, 0)\) is the correct x-intercept from the provided list.

Examples

In navigation, if a ship's path is modeled by a function \(f(t)\) representing its height relative to sea level at time \(t\), the x-intercepts (or t-intercepts) represent the moments when the ship is exactly at sea level. By checking a data table of the ship's altitude at different times, one can easily identify these crucial moments by looking for entries where the altitude is zero. This simple identification helps in planning maritime activities and ensuring safety.

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