Problem 42 - Algebra 2

Question

What is \(g(4)\) if \(g(x)=x^{2}-5 ?\)

Step-by-Step Solution

Answer

The value of \( g(4) \) is 11.

Step 1: Understand the Function

We are given a function \( g(x) = x^2 - 5 \). The function takes an input \( x \), squares it, and then subtracts 5 from the result. Our task is to find the value of the function when \( x = 4 \).

Step 2: Substitute the Value of x

To find \( g(4) \), substitute \( 4 \) into the function in place of \( x \). This gives us: \( g(4) = 4^2 - 5 \).

Step 3: Calculate the Square

Calculate \( 4^2 \), which is \( 16 \). So now the expression becomes: \( g(4) = 16 - 5 \).

Step 4: Subtract 5

Subtract 5 from the result obtained: \( 16 - 5 = 11 \).

Step 5: Result

The value of \( g(4) \) is \( 11 \).

Key Concepts

Quadratic Functions

Quadratic functions are fundamental in algebra and are characterized by the presence of a squared variable. A standard quadratic function is typically expressed in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The simplest form of a quadratic function, like \( g(x) = x^2 - 5 \), focuses on the squared term without a linear middle term or constant addition other than subtraction.Key features of quadratic functions include:

• The graph is a parabola, which is a U-shaped curve. This can either open upwards or downwards depending on the sign of the leading coefficient \( a \).
• The vertex, which is the highest or lowest point on the graph, occurs at the axis of symmetry of the parabola.
• These functions have a domain of all real numbers. However, their range depends on the direction the parabola opens—only non-negative values for an upward parabola starting from the vertex, for example.

In our example, since we have the function \( g(x) = x^2 - 5 \), it opens upwards because the coefficient of \( x^2 \) is positive (\( 1 \)), and its vertex lies along the y-axis.

Substitution Method

The substitution method is a basic but crucial technique in algebra where you replace a variable with a given number or another expression to simplify or solve an equation. This method is especially useful when evaluating functions, like finding \( g(4) \) for a given function \( g(x) \).Here's how the substitution method works:

• Start by identifying the variable you need to replace—in our case, \( x \) in the function \( g(x) = x^2 - 5 \).
• Replace \( x \) with the specified value—in the exercise, this is \( 4 \).
• Perform the arithmetic operations with the substituted value, which involves calculating \( 4^2 \, \text{in} \, g(x) = 4^2 - 5 \).
• Complete the calculation to find the value, resulting in \( g(4) = 11 \).

This method is straightforward and powerful, as it turns algebraic expressions into concrete numbers, paving the way for further analysis or graph plotting.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In the context of quadratic functions like \( g(x) = x^2 - 5 \), understanding algebraic expressions is necessary to manipulate and evaluate them.When dealing with algebraic expressions, consider the following:

• Identify the terms: Here, \( x^2 \) and \(-5\) are terms. The expression comprises both an exponential term and a constant.
• Simplification involves performing operations to reduce the expression to its simplest form. Each operation should be stepwise to avoid mistakes.
• Algebraic manipulation is also about following arithmetic rules, like the order of operations—performing powers before subtraction in expressions like \( 4^2 - 5 \).

By mastering algebraic expressions, you learn how to efficiently work through substitution and simplification, fundamental skills for solving many mathematical problems.