Problem 31 - College Algebra Essentials

Question

Evaluate each function at the given values of the independent variable and simplify. $$ h(x)=x^{4}-x^{2}+1 $$ a. \(h(2)\) b. \(h(-1)\) c. \(h(-x)\) d. \(h(3 a)\)

Step-by-Step Solution

Answer

The evaluations of the function at the given values are: a. \(h(2)=13\), b. \(h(-1)=1\), c. \(h(-x) = x^{4} - x^{2} + 1\), d. \(h(3a) = 81a^{4} - 9a^{2} + 1\).

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

To evaluate \(h(2)\), substitute \(x\) with 2 in the function: \(h(2) = (2)^{4} - (2)^{2} + 1 = 16 - 4 + 1 = 13\

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

Next, to evaluate \(h(-1)\), substitute \(x\) with -1 in the function: \(h(-1) = ((-1)^{4}) - ((-1)^{2}) + 1 = 1 - 1 + 1 = 1\)

Step 3: Evaluate \(h(-x)\)

Thirdly, to evaluate \(h(-x)\), substitute \(x\) with \(-x\) in the function, this implies using both the commutative property of addition and the even property of \(x^{2}\): \(h(-x) = ((-x)^{4}) - ((-x)^{2}) + 1 = x^{4} - x^{2} + 1\)

Step 4: Evaluate \(h(3a)\)

Finally, to evaluate \(h(3a)\), substitute \(x\) with \(3a\) in the function: \(h(3a) = ((3a)^{4}) - ((3a)^{2}) + 1 = 81a^{4} - 9a^{2} + 1\)

Key Concepts

Evaluating Functions

Evaluating a function essentially means replacing its variable with a specific value and calculating the result. This is crucial since it helps determine how a function behaves for particular inputs. Imagine a function as a machine where you insert a value and get an outcome. For example, given a function \( h(x) = x^4 - x^2 + 1 \), putting in the value 2, results in \( h(2) = 13 \). Here, the input 2 was substituted for \( x \), and the arithmetic was calculated accordingly.

• First, replace the variable with the given value.
• Then, follow the order of operations: exponents, multiplication, and addition/subtraction.
• Finally, simplify the expression to get the function's output for that specific input.

Experimenting with different values gives insights into how the function behaves and what kind of outputs it can produce.

Independent Variable

The independent variable in algebraic functions is the placeholder or symbol that represents the input value. In the context of functions, the independent variable typically determines the value of the output.
For our function \( h(x) = x^4 - x^2 + 1 \), \( x \) is the independent variable. It controls the function's behavior since changing \( x \) changes the value of \( h(x) \).

• Think of it as the cause in a cause-effect relationship; in mathematics, what we input is the independent variable, and the effect is seen on the dependent variable.
• It allows the function to be flexible, taking on infinite values depending on what we supply as \( x \).

Understanding independent variables helps in studying patterns within the function's results as the input changes.

Polynomial Expressions

Polynomial expressions are a type of algebraic expression that involve sums of powers of variables. These powers are always non-negative integers. In simpler terms, they are expressions formed by adding together multiple terms, each term being a product of a coefficient and a variable raised to an exponent.
The function \( h(x) = x^4 - x^2 + 1 \) is a polynomial. It is composed of three terms: \( x^4 \), \( -x^2 \), and \( +1 \). Here:

• The highest power of \( x \) is 4, which means this polynomial is of degree 4.
• Each term involves a single variable, raised to a power, and potentially multiplied by a coefficient (though in this case, those coefficients are 1, -1, and 1).
• Constant terms, like \( +1 \) in our function, are also considered part of a polynomial.

Grasping polynomial expressions is foundational for analyzing and solving more complex algebraic equations and functions.