Question

Sketch the graph of \(F(t)=\frac{|t|-t}{t}\)

Short answer

The graph consists of two horizontal lines: \(y = 0\) for \(t > 0\) and \(y = -2\) for \(t < 0\), with a discontinuity at \(t = 0\).

Step-by-step solution

Identify the Function Components

The function is given as \( F(t) = \frac{|t|-t}{t} \). We need to recognize that this function involves the absolute value \(|t|\) and a subtraction from \(t\). The function will behave differently depending on whether \(t\) is positive, negative, or zero.

Divide into Cases Based on t

Let's consider different cases for the domain of \(t\): - Case 1: \(t > 0\)- Case 2: \(t < 0\)- Case 3: \(t = 0\)

Analyze Case 1 (t > 0)

For \(t > 0\), we know that \(|t| = t\). Substitute into the function:\[ F(t) = \frac{t - t}{t} = \frac{0}{t} = 0 \]Therefore, when \(t > 0\), \(F(t) = 0\).

Analyze Case 2 (t < 0)

For \(t < 0\), the absolute value \(|t| = -t\) because \(t\) is negative. Substitute into the function:\[ F(t) = \frac{-t - t}{t} = \frac{-2t}{t} = -2 \]Therefore, when \(t < 0\), \(F(t) = -2\).

Analyze Case 3 (t = 0)

For \(t = 0\), the function \(F(t)\) is undefined because we cannot divide by zero. Thus, \(F(t)\) does not exist at \(t = 0\).

Combine Results to Sketch the Graph

Based on the cases analyzed:- For \(t > 0\), the graph of \(F(t)\) is a horizontal line at \(y=0\).- For \(t < 0\), the graph of \(F(t)\) is a horizontal line at \(y=-2\).- At \(t = 0\), there is a point of discontinuity.When sketching, illustrate the horizontal lines for \(t > 0\) and \(t < 0\) with an open circle at \(t = 0\) to show the discontinuity.

Key concepts

Absolute Value Function

An absolute value function involves the absolute value of a number, which is always non-negative. The absolute value, denoted by \(|t|\), represents the distance of a number from zero on the number line. This distance is always positive, regardless of whether the original number is negative or positive. This property is crucial because it affects how functions behave based on the sign of their variable.

- For positive numbers, the absolute value is the number itself: - \(|t| = t\) if \(t > 0\).- For negative numbers, the absolute value negates the number to make it positive: - \(|t| = -t\) if \(t < 0\).- The absolute value of zero is zero: - \(|t| = 0\).
In the context of the given function \(F(t) = \frac{|t| - t}{t}\), this means that the function's behavior changes depending on whether \(t\) is positive, negative, or zero. The key takeaway is understanding how absolute value manipulates the variable \(t\) into non-negative terms for different ranges.

Piecewise Functions

Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the main function's domain. They are useful for describing situations in which different rules apply in different cases.

In the function \(F(t) = \frac{|t| - t}{t}\), \(F(t)\) can be considered a piecewise function:

• For \(t > 0\), we substitute \(|t|\) with \(t\), resulting in \(F(t) = 0\).
• For \(t < 0\), we substitute \(|t|\) with \(-t\), resulting in \(F(t) = -2\).
• At \(t = 0\), the function is undefined because division by zero is not possible.

This function displays different behaviors depending on the value of \(t\). The piecewise nature is evident because the rules for calculating outputs change with the sign of \(t\). Understanding piecewise functions is key to handling cases where a single formula cannot describe a function's entire behavior.

Discontinuity in Functions

Discontinuity in a function occurs when a function is not defined at a certain point or when there is a sudden break in the behavior of the graph.

For the function \(F(t) = \frac{|t| - t}{t}\), there is a discontinuity at \(t = 0\). At this point,
- Division by zero occurs, making \(F(t)\) undefined at \(t = 0\).
This creates a gap or an open circle in the graph, indicating that the function does not have a continuous value at that point. Discontinuity is important to recognize because it affects the integrity of the function across its domain and alters how the function is graphed. For students, understanding discontinuity helps identify and handle exceptions in function behavior efficiently.

Horizontal Lines

Horizontal lines in the graph of a function signify that the function has a constant output over a particular interval. The slope of a horizontal line is zero:
- For the function \(F(t) = \frac{|t| - t}{t}\), a horizontal line is present when the output value remains the same for intervals of \(t\).

• If \(t > 0\), \(F(t)\) equals zero. So, the graph is a horizontal line along the \(y = 0\) axis.
• If \(t < 0\), \(F(t)\) equals \(-2\). Hence, the graph is a horizontal line at \(y = -2\).
• There's an open circle at \(t = 0\), indicating the discontinuity.

Horizontal lines in graphs help simplify analysis because they highlight constant values over specific intervals, providing visual clarity about the function's behavior in those ranges.