Chapter 2: Problem 28
Decide whether the function is even, odd, or neither. \(f(t)=t^{2}+3 t-10\)
Short Answer
Expert verified
The function \(f(t)=t^{2}+3 t-10\) is neither even nor odd.
Step by step solution
01
Estimate the function for evenness
Substitute \(-t\) for \(t\) in the function \(f(t)\), resulting in \(f(-t)=( -t)^{2}+3( -t)-10\). Simplify this expression to get \((-t)^{2}-3t-10\). Compare this to the original function \(f(t)=t^{2}+3 t-10\). If they were equal, it would be an even function.
02
Estimate the function for oddness
Now, check if the negative of the original function equal to \(f(-t)\), that is check if \(-f(t)=-t^{2}-3t+10\) equals \((-t)^{2}-3t-10\). If they were equal, it would be an odd function.
03
Conclude on function type
Since neither the condition of evenness nor the condition of oddness is satisfied, we can conclude that the function \(f(t)=t^{2}+3 t-10\) is neither even nor odd.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetry of Functions
Understanding the symmetry of functions is crucial when analyzing mathematical functions, as it reveals whether a function is even, odd, or neither. Even functions exhibit symmetry about the y-axis, meaning that for every point on the function, there is a mirrored point across the y-axis. The algebraic test for an even function is that if you replace every instance of the variable with its negative, the function remains unchanged, formally expressed as \( f(t) = f(-t) \).
Odd functions, on the other hand, show rotational symmetry about the origin. This symmetry indicates that if a point on the function exists at \( (t, f(t)) \), then another point exists at \( (-t, -f(t)) \). The algebraic test is that the function evaluated at the negation of the variable equals the negation of the function, or \( f(-t) = -f(t) \).
For the function \( f(t)=t^{2}+3t-10 \), we attempted to find such symmetries but ended up failing both conditions for evenness and oddness. This implies that the function has no symmetry about the y-axis or the origin, categorizing it as neither even nor odd. The lack of symmetry helps us understand the behavior of this function and implies that predictions based on symmetry cannot be made for values of \( t \).
Odd functions, on the other hand, show rotational symmetry about the origin. This symmetry indicates that if a point on the function exists at \( (t, f(t)) \), then another point exists at \( (-t, -f(t)) \). The algebraic test is that the function evaluated at the negation of the variable equals the negation of the function, or \( f(-t) = -f(t) \).
For the function \( f(t)=t^{2}+3t-10 \), we attempted to find such symmetries but ended up failing both conditions for evenness and oddness. This implies that the function has no symmetry about the y-axis or the origin, categorizing it as neither even nor odd. The lack of symmetry helps us understand the behavior of this function and implies that predictions based on symmetry cannot be made for values of \( t \).
Algebraic Functions
Algebraic functions are functions that can be expressed using algebraic operations, including addition, subtraction, multiplication, division, and taking roots within a finite number of steps. The function presented in our exercise, \( f(t)=t^{2}+3t-10 \), is an example of an algebraic function as it is built from polynomial expressions.
An important feature of algebraic functions is their equation's degree, which is determined by the highest power of the variable within the equation. In our case, the degree is 2, since the highest power of \( t \) is in the term \( t^2 \). The degree can give insights into the potential symmetry of the function. For instance, simple polynomial functions with only even powers are even functions, and those with only odd powers are odd functions. However, our function mixes even and odd powers and thus eludes a clear classification based on symmetry.
When analyzing algebraic functions, we often look at their graphs, roots, and behavior as the variable approaches infinity or negative infinity. In the classroom or for homework exercises, identifying whether these functions are even or odd, like in our example, is a fundamental skill that assists in understanding their graph and behavior.
An important feature of algebraic functions is their equation's degree, which is determined by the highest power of the variable within the equation. In our case, the degree is 2, since the highest power of \( t \) is in the term \( t^2 \). The degree can give insights into the potential symmetry of the function. For instance, simple polynomial functions with only even powers are even functions, and those with only odd powers are odd functions. However, our function mixes even and odd powers and thus eludes a clear classification based on symmetry.
When analyzing algebraic functions, we often look at their graphs, roots, and behavior as the variable approaches infinity or negative infinity. In the classroom or for homework exercises, identifying whether these functions are even or odd, like in our example, is a fundamental skill that assists in understanding their graph and behavior.
Function Analysis
Function analysis involves examining various attributes of functions such as domain, range, symmetry, intercepts, and behavior at extremities. It helps us predict the function's graph and apply it to solve practical problems. In the case of the exercise function \( f(t)=t^{2}+3t-10 \), function analysis starts with identifying potential symmetry, as we earlier discussed, and then proceeds to other characteristics.
For instance, the intercepts of our function can be found by setting the function equal to zero and solving for \( t \), which would give us the x-intercepts or roots. The function's behavior as \( t \) approaches infinity or negative infinity can indicate end behavior, helping sketch the graph's tail ends. Moreover, examining the function's derivative can reveal information about the slope of the function and potential maxima and minima points.
It's important for students to realize that function analysis is not merely about finding if the function is even or odd, but it’s a holistic approach to understanding every aspect of the function's behavior. This broader analysis is essential for higher-level mathematics and applications in scientific fields.
For instance, the intercepts of our function can be found by setting the function equal to zero and solving for \( t \), which would give us the x-intercepts or roots. The function's behavior as \( t \) approaches infinity or negative infinity can indicate end behavior, helping sketch the graph's tail ends. Moreover, examining the function's derivative can reveal information about the slope of the function and potential maxima and minima points.
It's important for students to realize that function analysis is not merely about finding if the function is even or odd, but it’s a holistic approach to understanding every aspect of the function's behavior. This broader analysis is essential for higher-level mathematics and applications in scientific fields.
