Problem 40 - Thinking Mathematically

Question

Evaluate each function at the given value of the variable. \(g(x)=-x^{2}+1\) a. \(g(5)\) b. \(g(-4)\)

Step-by-Step Solution

Answer

\(g(5) = -24\) and \(g(-4) = -15\)

Step 1: Evaluation for \(x=5\)

Substitute \(x = 5\) into the function \(g(x)\), so the function becomes \(g(5) = -(5)^{2} + 1\). Now, perform the squaring operation first (which results in \(25\)), then apply the unary operation of negation (which yields \(-25\)), and finally, add \(1\).

Step 2: Simplification for \(x=5\)

After performing all operations, the final result is \(g(5) = -24\).

Step 3: Evaluation for \(x=-4\)

Now substitute \(x = -4\) into the function \(g(x)\), so the function becomes \(g(-4) = -(-4)^{2} + 1\). As before, perform the squaring operation first (resulting in \(16\)), then apply unary negation (giving \(-16\)), and finally, add \(1\).

Step 4: Simplification for \(x=-4\)

After simplification, the final result is \(g(-4) = -15\).

Key Concepts

Unary Operation

Unary operations are mathematical operations that involve only one operand. In simpler terms, you only need one number or variable to perform the operation. One common unary operation you might encounter is the negation operation. This involves changing the sign of a number. For example:

• If you have a positive number, applying negation makes it negative.
• For a negative number, negation will make it positive.

In the function given in the exercise, the unary operation is used to negate the result after squaring. Understanding unary operations helps simplify expressions by making it clear when and how to invert signs.

Squaring Operation

Squaring a number means multiplying the number by itself. This operation is represented by raising the number to the power of 2. In mathematical notation, we write it as:

• \[ x^2 = x \times x \]
• For example, when you square 5, you calculate \(5^2 = 25\).

The squaring operation is significant because it is the first step in evaluating the given function: first, square the input value, then apply other operations. Squaring always results in a non-negative number, which is crucial when further operations, such as negation, are applied later. Understanding the squaring operation ensures that calculations are performed accurately from the outset.

Substitution

Substitution is a technique used to evaluate expressions. You replace a variable with a specific value to simplify or solve an equation. In the context of functions:

• You replace the variable \(x\) with a specific number, called the argument.
• This allows you to calculate the function's value at that specific point.
• For example, substituting \(x = 5\) in \(g(x) = -x^2 + 1\), you perform operations on \(g(5)\) instead of dealing with the general variable \(x\).

Substitution is a crucial step in function evaluation, as it provides the specific outputs based on different inputs, helping describe how the function behaves across its domain.

Polynomials

Polynomials are algebraic expressions composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. A polynomial function might look like:

• \[ g(x) = -x^2 + 1 \]
• Here, \(-x^2\) and \(+1\) are terms of the polynomial.

Polynomials can range from simple linear expressions (like \(ax + b\)) to much more complex forms. The highest exponent of the variable in a polynomial determines its degree, which is 2 in the given function, making it a quadratic polynomial.
In mathematical functions, polynomials serve as foundations for more complex calculations and transformations, providing essential knowledge for graphing and solving equations.