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Chapter 14: Problem 44

Sketch the graph of each function. Decide whether each function is one-to-one. \(F(x)=-2\)

Short Answer

Expert verified
The graph is a horizontal line at \\(y = -2\\\); it is not one-to-one.

Step by step solution

01

Identify the Function Type

The given function is a constant function, because it is defined by a single value, which is \( -2 \). It does not change with different values of \( x \).
02

Sketch the Graph

To sketch the graph of the function \(F(x) = -2\), draw a horizontal line at \(y = -2\) across the entire coordinate plane. This line remains constant for all values of \(x\).
03

Determine if the Function is One-to-One

A function is one-to-one if each output value \(y\) is uniquely associated with only one input value \(x\). For \(F(x) = -2\), all inputs \( x \) produce the same output \( -2 \), so it is not one-to-one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing a function involves creating a visual representation of the function on a coordinate plane. This helps us see the relationship between the input (usually on the x-axis) and the output (usually on the y-axis). For constant functions like \(F(x) = -2\), the graph is a straight line. No matter what value \(x\) takes, the corresponding \(y\)-value remains \(-2\).
  • A constant function graph is always parallel to the x-axis.
  • To draw this graph, simply place a horizontal line at the y-value of the constant, here \(-2\).
  • This line continues infinitely in both directions along the x-axis.
Graphing functions not only helps in understanding but also in analyzing the function's properties, like intercepts, increases, or decreases in value.
One-to-One Functions
A one-to-one function is a special type of function where each value of \(y\) that results from the function is paired with a unique \(x\) value. In other words, no two different \(x\) values should map to the same \(y\) value.
  • A simple way to check if a function is one-to-one is by using the Horizontal Line Test.
  • If any horizontal line crosses the graph of the function more than once, the function is not one-to-one.
  • For our constant function, \(F(x) = -2\), a horizontal line across the graph at \(y = -2\) will touch it infinitely many times since the entire graph is at \(y = -2\).
Hence, a constant function can never be one-to-one because the definition inherently clashes: multiple \(x\) values give the same \(y\) value.
Horizontal Line
The term 'horizontal line' refers to a straight line in a graph that runs parallel to the x-axis. Such lines have the same y-value across all points, which is why they form the graphs of constant functions. In our example, the horizontal line is at \(y = -2\).
  • The equation of a horizontal line can be expressed as \(y = c\), where \(c\) is the constant value the line lies on.
  • Horizontal lines depict scenarios where changes in \(x\) do not affect \(y\), exemplifying constant functions.
  • These lines are critical in determining one-to-one characteristics via the Horizontal Line Test.
Understanding horizontal lines is essential for graphing and interpreting constant functions and other related mathematical analyses.

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Most popular questions from this chapter

Find the sum of the terms of each infinite geometric sequence. $$-3, \frac{3}{5},-\frac{3}{25}, \ldots$$

Find the sum of the terms of each infinite geometric sequence. $$\frac{1}{10}, \frac{1}{100}, \frac{1}{1000}, \dots$$

Given are the first three terms of a sequence that is either arithmetic or geometric If the sequence is arithmetic, find \(a_{1}\) and \(d\). If a sequence is geometric, find \(a_{1}\) and \(\bar{r}\) $$ t, t-1, t-2 $$

Write the first four terms of the arithmetic or geometric sequence whose first term, \(a_{1},\) and common difference, \(d\), or common ratio, \(r\) are given. $$ a_{1}=19.652 ; d=-0.034 $$

Solve. If \(a_{1}\) is \(10, a_{18}\) is \(\frac{3}{2},\) and \(d\) is \(-\frac{1}{2},\) find \(S_{18}\)
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