Problem 38 - Thinking Mathematically

Question

Evaluate each function at the given value of the variable. \(g(x)=x^{2}+4\) a. \(g(3)\) b. \(g(-3)\)

Step-by-Step Solution

Answer

The solution is \(g(3) = 13\) and \(g(-3) = 13\).

Step 1: Substitute x = 3

Firstly, substitute \(x = 3\) into the equation: \(g(3) = (3)^{2} + 4\). Do the calculation for \((3)^{2} = 9\), this will give \(g(3) = 9 + 4\).

Step 2: Calculate g(3)

Then, perform the addition operation to get the value: \(g(3) = 9 + 4 = 13\).

Step 3: Substitute x = -3

Next, substitute \(x = -3\) into the equation: \(g(-3) = (-3)^{2} + 4\). Perform the calculation for \((-3)^{2} = 9\). this gives \(g(-3) = 9 + 4\).

Step 4: Calculate g(-3)

Then, perform the addition operation to get the value: \(g(-3) = 9 + 4 = 13\).

Key Concepts

Quadratic Functions

A quadratic function is a type of mathematical expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The key characteristic of a quadratic function is that the highest exponent of the variable \(x\) is 2.Quadratic functions are graphically represented as parabolas. Depending on the coefficient \(a\), the parabola may open upwards or downwards:

• If \(a > 0\), the parabola opens upwards, resembling a 'U' shape.
• If \(a < 0\), the parabola opens downwards, resembling an upside-down 'U'.

In the function \(g(x)=x^2+4\), since the coefficient of \(x^2\) is positive 1, the parabola opens upwards.Understanding how to evaluate such functions at specific values helps in analyzing their behavior and application in real-world problems.

Substitution Method

The substitution method is a straightforward technique used to determine the value of a function at a specific point. To apply this method, you replace the variable \(x\) with a given number.For example, if you need to find \(g(3)\) in the function \(g(x)=x^2+4\):

• Replace \(x\) with 3 in the expression, resulting in \(g(3)=(3)^2+4\).
• Compute \((3)^2\) which equals 9.
• Add the constant 4, giving you \(g(3)=13\).

Similarly, for \(g(-3)\), substitute \(x\) with -3:

• This results in \(g(-3)=(-3)^2+4\).
• Calculate \((-3)^2\), which is also 9.
• Add the 4 to get \(g(-3)=13\).

This method is key to evaluating functions and understanding how their outputs vary with different inputs.

Algebraic Expressions

Algebraic expressions like \(x^2 + 4\) in the given function \(g(x)\) consist of numbers, variables, and operators. These expressions can model real-world scenarios and provide a foundation for algebraic problem-solving.In the given algebraic expression:

• \(x^2\) is a term representing the square of the variable \(x\).
• The number 4 is a constant term that does not change regardless of \(x\).
• Adding these two components together forms the complete expression \(x^2 + 4\).

When evaluating such expressions through substitution, it entails inserting particular values into the variable place.These evaluations offer insights into the numerical value of the expression at that point. Grasping how to manipulate and evaluate algebraic expressions is an essential skill in math, useful in various scientific and engineering applications.