Problem 17
Question
Sketch the graph of the function. $$ f(t)=|t|-t $$
Step-by-Step Solution
Verified Answer
The graph is a horizontal line at 0 for \( t \geq 0 \) and a line with slope -2 for \( t < 0 \).
1Step 1: Analyze the Function
The given function is \( f(t) = |t| - t \). The absolute value function \( |t| \) outputs \( t \) when \( t \geq 0 \) and \( -t \) when \( t < 0 \). Let's analyze these two cases separately.
2Step 2: Case Analysis for \( t \geq 0 \)
For \( t \geq 0 \), we have \( |t| = t \). Therefore, the function simplifies to: \[ f(t) = t - t = 0. \]So for any \( t \geq 0 \), the function value is 0.
3Step 3: Case Analysis for \( t < 0 \)
For \( t < 0 \), \( |t| = -t \). The function becomes: \[ f(t) = -t - t = -2t. \] This indicates a linear graph with a slope of -2 for \( t < 0 \).
4Step 4: Sketch the Graph
To sketch the graph, observe the results from the previous steps:- For \( t \geq 0 \), draw a horizontal line at \( y = 0 \).- For \( t < 0 \), draw a line with a slope of -2 starting from the origin, heading downward as it moves left.The graph will be a horizontal line for non-negative values and a straight line with slope -2 for negative values.
Key Concepts
Piecewise FunctionsAbsolute Value FunctionLinear Function
Piecewise Functions
Piecewise functions are fascinating and versatile because they allow a function to behave differently over different parts of its domain. In the given exercise, the function is defined separately for different intervals of the variable \( t \). Specifically, it separates behavior at \( t=0 \).
- When dealing with piecewise functions, it is crucial to evaluate the conditions or intervals over which different expressions apply. This graph of \( f(t) = |t| - t \) behaves differently for \( t \geq 0 \) and \( t < 0 \). - For \( t \geq 0 \), the value of \( f(t) \) is constant, resulting in a horizontal line part on the graph. - For \( t < 0 \), we see a linear expression, forming a downward sloping line due to the negative sign.
The essence of a piecewise function is its ability to seamlessly draw from different function types to fit a problem space. Understanding these can help you make sense of graphs that aren't straightforward and appreciate the diversity in mathematical modeling.
- When dealing with piecewise functions, it is crucial to evaluate the conditions or intervals over which different expressions apply. This graph of \( f(t) = |t| - t \) behaves differently for \( t \geq 0 \) and \( t < 0 \). - For \( t \geq 0 \), the value of \( f(t) \) is constant, resulting in a horizontal line part on the graph. - For \( t < 0 \), we see a linear expression, forming a downward sloping line due to the negative sign.
The essence of a piecewise function is its ability to seamlessly draw from different function types to fit a problem space. Understanding these can help you make sense of graphs that aren't straightforward and appreciate the diversity in mathematical modeling.
Absolute Value Function
The absolute value function \( |t| \) plays a central role in the problem, characterized by returning the non-negative magnitude of \( t \), regardless of its sign. This mathematical tool checks whether a number is positive or negative, and then either returns it as it is or as its positive counterpart respectively.
- For \( t \geq 0 \), \( |t| = t \), which means if \( t \) is positive or zero, the function remains unaffected. This simplification results in the output of our piecewise function as \( f(t) = 0 \).- On the contrary, when \( t < 0 \), \( |t| \) turns to \( -t \), effectively flipping the value of the input to positive. This leads the sub-function \( f(t) \) to turn into \( -2t \), a linear expression with a descending perspective.
Absolute value functions are essential in handling real number problems, especially where the direction or sign of a value doesn't alter the ongoing analysis but only its magnitude is necessary.
- For \( t \geq 0 \), \( |t| = t \), which means if \( t \) is positive or zero, the function remains unaffected. This simplification results in the output of our piecewise function as \( f(t) = 0 \).- On the contrary, when \( t < 0 \), \( |t| \) turns to \( -t \), effectively flipping the value of the input to positive. This leads the sub-function \( f(t) \) to turn into \( -2t \), a linear expression with a descending perspective.
Absolute value functions are essential in handling real number problems, especially where the direction or sign of a value doesn't alter the ongoing analysis but only its magnitude is necessary.
Linear Function
The linear function segment of our exercise emerges in the interval where \( t < 0 \). When simplified, the original function \( f(t) = -2t \) showcases the characteristics of a straight line.
- A key feature of linear functions like \( -2t \) is that they produce a constant rate of change or a slope. In this instance, the slope is \(-2\), indicating that for every unit decrease in \( t \) (to the left on the graph), the function \( f(t) \) decreases by 2 units.
- Linear functions are plotted as straight lines on a graph, distinguished usually by their slope and y-intercept. However, in this expression \(-2t\), the slope is emphasized over the intercept since it crosses the origin when \( t=0 \).
Understanding linear functions is crucial for graph interpretation and creation, as they form the basis for prediction and trend analysis models in both pure and applied mathematical areas.
- A key feature of linear functions like \( -2t \) is that they produce a constant rate of change or a slope. In this instance, the slope is \(-2\), indicating that for every unit decrease in \( t \) (to the left on the graph), the function \( f(t) \) decreases by 2 units.
- Linear functions are plotted as straight lines on a graph, distinguished usually by their slope and y-intercept. However, in this expression \(-2t\), the slope is emphasized over the intercept since it crosses the origin when \( t=0 \).
Understanding linear functions is crucial for graph interpretation and creation, as they form the basis for prediction and trend analysis models in both pure and applied mathematical areas.
Other exercises in this chapter
Problem 17
Sketch the graph. List the intercepts and describe the symmetry (if any) of the graph. $$ y=\frac{1}{3} x $$
View solution Problem 17
Write the union of the two intervals as a single interval. $$ (1,3) \text { and }(2, \infty) $$
View solution Problem 17
Find the domain of the function. $$ k(x)=1+x^{3} \text { for }-2 \leq x \leq 8 $$
View solution Problem 17
Find the domains and rules of \(f+g, f g\), and \(f / g .\) \(f(t)=t^{3 / 4}\) and \(g(t)=t^{2}+3\)
View solution