Problem 42 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \(h(t) = t^2-2t\) (a) \(h(2)\) (b) \(h(1.2)\) (c) \(h(x+2)\)

Step-by-Step Solution

Answer

The simplified values of the function \(h(t) = t^2 - 2t\) are \(h(2) = 0\), \(h(1.2) = -0.96\), and \(h(x+2) = x^2 + 2x\).

Step 1: Evaluate \(h(2)\)

Substitute \(t = 2\) in the function \(h(t) = t^2 - 2t\). This gives us \(h(2) = 2^2 - 2*2 = 4 - 4 = 0\)

Step 2: Evaluate \(h(1.2)\)

Now substitute \(t = 1.2\) in the function \(h(t) = t^2 - 2t\). Therefore, \(h(1.2) = (1.2)^2 - 2*1.2 = 1.44 - 2.4 = -0.96\)

Step 3: Evaluate \(h(x+2)\)

Finally, substitute \(t = x+2\) in the function \(h(t) = t^2 - 2t\). Simplify to get \(h(x+2) = (x+2)^2 - 2*(x+2) = x^2 + 4x + 4 - 2x - 4 = x^2 + 2x\)

Key Concepts

Understanding Independent Variables

In the realm of functions, the term "independent variable" is foundational. An independent variable is the input of a function, which you have the freedom to choose. It is typically represented by a letter such as \(t\) or \(x\). It serves as a placeholder that determines the output or the dependent variable.
For instance, in the function \(h(t) = t^2 - 2t\), \(t\) is the independent variable. The value of \(h(t)\) changes depending on the number you substitute for \(t\).
Independent variables are crucial as they allow functions to model real-world scenarios, showing how changing one factor affects another. Whenever you evaluate a function at a specific point, you are essentially selecting a value for your independent variable and observing the resulting output.

Mastering the Substitution Method

The substitution method is a handy technique when evaluating functions. This involves replacing the independent variable in the function with a specific value to find the output.
For example, to evaluate \(h(t) = t^2 - 2t\) at \(t = 2\), we substitute 2 into the function:

• Replace each instance of \(t\) with 2: \(h(2) = 2^2 - 2 \times 2\).
• Simplify the expression: \(h(2) = 4 - 4 = 0\).

This straightforward process makes substitution a powerful tool in mathematics. It allows for evaluating functions not only at specific numbers but also at expressions, enhancing the understanding of how functions behave.
When using substitution, ensure you carefully replace each occurrence of the independent variable to avoid errors in calculation.

Exploring Polynomial Functions

Polynomial functions form a broad and vital class of mathematical expressions. They consist of variables raised to whole-number exponents, multiplied by coefficients, and potentially combined with constant terms. A general polynomial might look like \(ax^n + bx^{n-1} + \ldots + k\).
The function \(h(t) = t^2 - 2t\) is a polynomial of degree 2, also known as a quadratic polynomial. The degree of a polynomial is determined by the highest power of the variable. In this case, the highest power is 2, thus it is quadratic.
Polynomial functions are highly versatile and applicable across various areas of science and engineering. They help model phenomena such as projectile motion or population growth. Understanding and working with polynomial functions involve recognizing their structure and rules for operations such as addition, subtraction, and multiplication.
Simplifying polynomial expressions by combining like terms, as seen in evaluating \(h(x+2)\), demonstrates their dynamic nature and the ease with which they can be manipulated.