Problem 10 - Basic Technical Mathematics with Calculus

Question

Find the domain and range of the given functions. $$T(t)=2 t^{4}+t^{2}-1$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \([-1, \infty)\).

Step 1: Determine the Domain

Since the function is a polynomial, we know that polynomials are defined for all real numbers. Hence, the domain of the function is all real numbers, i.e., \( \text{Domain of } T(t) = (-\infty, \infty) \).

Step 2: Determine the Behavior of the Function

To find the range, note that the leading term of the polynomial is \( 2t^4 \), which suggests that this is an even degree polynomial with a positive leading coefficient. This means as \( t \) approaches \( \infty \) or \( -\infty \), \( T(t) \) will approach \( \infty \).

Step 3: Identify the Minimum Value

Since polynomials of even degree have minimum values, we need to find the minimum of the function. We do this by finding the derivative, \( T'(t) = 8t^3 + 2t \), and setting it equal to zero to find critical points:\[ 8t^3 + 2t = 0 \]Factoring gives:\[ 2t(4t^2 + 1) = 0 \]The solutions are \( t = 0 \) and \( 4t^2 + 1 = 0 \). Since \( 4t^2 + 1 = 0 \) has no real solutions, the only critical point is \( t = 0 \).

Step 4: Evaluate the Function at Critical Points

Evaluating the function at \( t = 0 \) gives:\[ T(0) = 2(0)^4 + (0)^2 - 1 = -1 \].Since the polynomial tends to infinity as \( t \to \pm \infty \), the minimum value is \( -1 \).

Step 5: Determine the Range

Since the function has a minimum value of \( -1 \) and increases towards infinity, the range of the function is all values greater than or equal to \( -1 \):\( \text{Range of } T(t) = [-1, \infty) \).

Key Concepts

Domain and Range

When dealing with functions, the domain and range are fundamental concepts to understand. The domain refers to all the possible input values (usually represented as "x" or in this case, "t") that a function can accept. For a polynomial function like \(T(t) = 2t^4 + t^2 - 1\), there are no restrictions on "t" since you can plug any real number into the equation and compute a result.
Therefore, the domain of this polynomial function is all real numbers: \((-\infty, \infty)\).

• Domain: All real numbers
• Common notation: \((-\infty, \infty)\)
• No input restrictions

Next, let's explore the range, which describes all possible output values of the function. Understanding the leading term gives clues about the function's behavior. For \(T(t)\), as \(t\) becomes very large or very small, the output tends towards infinity due to the presence of the \(t^4\) term. However, the function has a minimum value, which we'll find by using calculus. Since \(T(t)\) can reach any output value above \(-1\), the range is \([-1, \infty)\). This means the function's smallest possible value is \(-1\), and it can grow arbitrarily large.

Critical Points

Critical points of a function are where the derivative is zero or undefined. These points often indicate local minimums, maximums, or other interesting behavior in the function. For our polynomial \(T(t) = 2t^4 + t^2 - 1\), to find critical points, we first need the derivative: \(T'(t) = 8t^3 + 2t\).
Setting this derivative equal to zero allows us to find where the slope of the tangent line to the graph of the function is zero:
\[8t^3 + 2t = 0\]
Factoring gives:
\[2t(4t^2 + 1) = 0\]
The solutions to this are \(t = 0\) (since \(4t^2 + 1 = 0\) has no real solutions).

• \(t = 0\) is a critical point where we can find information about the function's behavior.
• Finding and analyzing critical points helps determine local behavior, such as valleys or peaks.

Checking the function at \(t = 0\) yields \(T(0) = -1\), marking it as the function's minimum value.

Leading Coefficient

In a polynomial function, the leading coefficient is the coefficient of the term with the highest power. It plays a critical role in determining the end behavior of the polynomial. For \(T(t) = 2t^4 + t^2 - 1\), the leading coefficient is \(2\), associated with the highest degree term \(t^4\).

• The leading coefficient influences how the polynomial behaves as the input variable \(t\) becomes very large or very small.
• This coefficient, when positive, ensures the polynomial extends upwards as \(t\) approaches \(\pm \infty\). If it were negative, the polynomial would turn downward.

Knowing the leading coefficient helps anticipate whether the polynomial "opens" upward or downward. For this polynomial, since \(2\) is positive, \(T(t)\) opens upward, meaning both ends of the graph rise towards infinity.

Even Degree Polynomial

Polynomials can be characterized by their degree, which is determined by the highest power of the variable in the polynomial. When a polynomial has an even degree, it brings specific properties.

• The polynomial \(T(t) = 2t^4 + t^2 - 1\) is of fourth degree, which is even.
• Even degree polynomials with positive leading coefficients have graphs that rise in both directions as \(t\) goes to \(\pm \infty\).
• These polynomials often have a minimum (or if the leading coefficient were negative, a maximum) value.

Understanding the nature of even degree polynomials helps predict these rise and fall patterns. In our function, this trait explains why \(T(t)\) heads towards infinity as \(t\) moves away from zero in either direction.