Problem 56 - Elementary and Intermediate Algebra

Question

Find \(g(2)\) and \(g(3) .\) See Example 4. $$ g(x)=(x-3)^{2} $$

Step-by-Step Solution

Answer

\(g(2) = 1\) and \(g(3) = 0\).

Step 1: Understand the Function

Identify that we are given a function \( g(x) = (x - 3)^2 \). This is a quadratic function representing a parabola where \( x \) is any real number.

Step 2: Substitute for \( g(2) \)

Replace \( x \) with \( 2 \) in the function: \[g(2) = (2 - 3)^2\]

Step 3: Simplify \( g(2) \)

Calculate the expression by performing the subtraction first: \[2 - 3 = -1\]Then, square the result: \[(-1)^2 = 1\]Thus, \( g(2) = 1 \).

Step 4: Substitute for \( g(3) \)

Replace \( x \) with \( 3 \) in the function: \[g(3) = (3 - 3)^2\]

Step 5: Simplify \( g(3) \)

Perform the subtraction: \[3 - 3 = 0\]Then, square the result: \[0^2 = 0\]Thus, \( g(3) = 0 \).

Key Concepts

Function Evaluation

When evaluating a function, the process involves substituting the given input value into the function's formula. This is like inserting ingredients into a recipe and getting a result or output. For the function \( g(x) = (x - 3)^2 \), we are tasked with determining what \( g(x) \) equals for specific values of \( x \).

• Step 1: Identify your function and the values you need to evaluate. Here, we need to find \( g(2) \) and \( g(3) \).
• Step 2: Substitute these values into the equation in place of \( x \). So for \( g(2) \), we replace \( x \) with 2 leading to \( (2 - 3)^2 \), and for \( g(3) \), replace \( x \) with 3 resulting in \( (3 - 3)^2 \).
• Step 3: Calculate the expression to find the output for each substitution. This involves some basic arithmetic.

This approach simplifies the process of finding the output for any input in the given function. It is especially useful in mathematics when working with polynomial functions like quadratics.

Parabola

A parabola is a symmetrical, curved, U-shaped line on a graph, which is commonly associated with quadratic functions. Quadratic functions, like \( g(x) = (x - 3)^2 \), produce parabolas that open either upwards or downwards.Here, our quadratic function is written in vertex form \( (x - h)^2 + k \), where the point \( (h, k) \) is the vertex of the parabola.For our function \( g(x) = (x - 3)^2 \):

• The vertex is at the point \( (3, 0) \), making it the lowest point on the curve for an upward-facing parabola.
• The axis of symmetry is the vertical line that passes through the vertex, \( x = 3 \). It divides the parabola into two symmetrical halves.

Understanding how a parabola behaves is key to visualizing the graph of a quadratic function. You can expect symmetrical attributes and specific points like the vertex that direct how the curve is shaped.

Simplification

Simplification is the process by which we reduce an expression to its simplest form, making calculations and interpretations much easier. In the case of evaluating \( g(x) = (x - 3)^2 \), simplification plays a crucial role in solving for \( g(2) \) and \( g(3) \).Here’s how simplification works in our example:

• Start with substitution: Replace \( x \) with the required value, like 2 or 3.
• Perform arithmetic operations: Begin with operations inside the brackets. For \( g(2) \), simplify \( (2-3)^2 \) into \( (-1)^2 \) and for \( g(3) \), \( (3-3)^2 \) into \( 0^2 \).
• Solve remaining expressions: Continue by calculating the power. For \( (-1)^2 \), it becomes 1, and for \( 0^2 \), it becomes 0.

By simplifying in steps, complex expressions become manageable. This is an essential skill in mathematics because it makes understanding and working with functions more straightforward.