Problem 37 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The function evaluations are: \(g(2)=5\) and \(g(-2)=5\).

Step 1: Substitute the Value Into Function for g(2)

Substitute the value \(x=2\) into the given function \(g(x)=x^{2}+1\). This will give a new equation \(g(2)=2^{2}+1\).

Step 2: Evaluate the Function for g(2)

Evaluate the function \(g(2)=2^{2}+1\) by performing the square operation and the addition operation. This results in \(g(2)=4+1\), and the ultimate result is \(g(2)=5\).

Step 3: Substitute the Value Into Function for g(-2)

Repeat the process for \(x=-2\). Substitute \(x=-2\) into the function \(g(x)=x^{2}+1\) to get the equation \(g(-2)=(-2)^{2}+1\).

Step 4: Evaluate the Function for g(-2)

Evaluate the equation \(g(-2)=(-2)^{2}+1\) by squaring -2 and then adding 1, giving \(g(-2)=4+1\). The ultimate result of the evaluation is \(g(-2)=5\).

Key Concepts

Function Evaluation

When it comes to understanding mathematical functions, the concept of function evaluation stands at the forefront. It is the process of determining the output of a function based on a specific input. Think of a function as a machine: you feed it an input, and it spits out an output following a set of rules defined by the function's formula. For example, if you have the function \( g(x)=x^2+1 \), and you are asked to evaluate this function for \( x=2 \), you essentially replace every instance of \( x \) in the function with 2. This results in a new equation specific to this input, \( g(2)=2^2+1 \), which simplifies to \( g(2)=5 \). This process is crucial as it helps students understand the behavior of the function at particular points, which is fundamental in calculus, physics, and many other fields of science and mathematics.

Evaluating a function involves a systematic approach. You identify the input, substitute it into the function's formula, and simplify the resulting expression to find the output. This helps students to not only work with abstract symbols but also to see how functions apply in real-world contexts where specific values are used.

Substitution Method

The substitution method plays a pivotal role in mathematics, especially when evaluating functions. It's a technique where you replace variables with numbers or other expressions. The methodology is straightforward: take the given input (like \( x=2 \) or \( x=-2 \)), and plug it into the equation wherever the variable appears. In our example with the function \( g(x)=x^2+1 \), you would substitute \( x \) with 2 to evaluate \( g(2) \), resulting in the equation \( g(2) = 2^2 + 1 \).

Why is Substitution Important?

• It reduces complex equations to simpler forms that are easier to evaluate.
• It lays the foundation for solving equations and understanding functions in a tangible way.
• It is a key skill in algebra and calculus for finding the precise value of expressions.

For students, mastering the substitution method means being able to navigate through a variety of mathematical problems with ease, translating abstract concepts into concrete numbers.

Square Operation

The square operation is another fundamental concept in mathematics. It refers to multiplying a number by itself. When you see an expression like \( x^2 \), it means \( x \) times \( x \). This is not to be confused with simply doubling a number. Doubling would mean \( 2x \), while squaring is equivalent to \( x \) multiplied by \( x \). When we evaluate \( g(x) = x^2 + 1 \) for \( x=2 \), the square operation comes into play as we compute \( 2^2 \), which is \( 2 \) times \( 2 \) and equals 4.

Understanding the Impact of Squaring

Squaring a number changes its value significantly. This operation is not commutative—meaning, the order in which you perform the squaring matters. Also, when squaring negative numbers, the result is always positive, as seen in \( g(-2) \), where \( (-2)^2 \) equals 4, the same result as squaring 2. Students benefit from practicing square operations, as they emerge in equations, functions, and graphing parabolas, therefore it is essential to grasp both the concept and its application.