Problem 2 - Algebra 2 and Trigonometry

Question

Let \(f(x)=x^{2}\) and \(g(x+2)=x^{2}+2 .\) Are \(f\) and \(g\) the same function? Explain why or why not.

Step-by-Step Solution

Answer

No, \(f\) and \(g\) are not the same function since they have different expressions.

Step 1: Understand the Problem Definitions

We are given two functions: \( f(x) = x^2 \) and \( g(x+2) = x^2 + 2 \). Our task is to determine if these two functions are identical. For this, we will need to compare their expressions and possibly perform substitutions to compare these functions at equivalent inputs.

Step 2: Rewrite g(x) with Standard Variable

The function \( g(x+2) = x^2 + 2 \) is given with an offset in its argument. We want to express it in terms of a typical \( x \) variable. Let us set \( u = x+2 \), then \( x = u-2 \). Substitute \( x = u-2 \) into the expression for \( g \): \( g(u) = (u-2)^2 + 2 \).

Step 3: Simplify g(u) to a Standard Form

Substitute \( x = u - 2 \) into the expression for \( g(x+2): \) \( g(u) = (u-2)^2 + 2 \). Expanding \( (u-2)^2 \) gives \( u^2 - 4u + 4 \), thus \( g(u) = u^2 - 4u + 4 + 2 = u^2 - 4u + 6 \).

Step 4: Compare f(x) and g(u)

Now, compare \( f(x) = x^2 \) with \( g(u) = u^2 - 4u + 6 \). For these to be the same function, the expressions must be identical, meaning they should have the same corresponding coefficients for equivalent polynomial terms.

Step 5: Conclude the Analysis

\( f(x) = x^2 \) has no linear term and a constant term of 0. \( g(u) = u^2 - 4u + 6 \) has additional linear and constant terms. Since both their linear and constant terms are not equal, \( f \) and \( g \) are not the same function.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. These functions are called "polynomial" due to the fact that the powers are positive integers. A simple example is the quadratic function, which is a polynomial of degree 2:

• In its standard form, a quadratic function is written as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\).
• The highest power of \(x\) in a quadratic polynomial is 2, making it a degree 2 polynomial.

In our original exercise, \(f(x) = x^2\) is a quadratic polynomial where \(a = 1\), \(b = 0\), and \(c = 0\). It is simple because it lacks a linear term (\(bx\)) and a constant term (\(c\)). This makes the comparison with other polynomials straightforward since any difference in the terms immediately indicates that they are not equivalent.
Understanding the structure of polynomial functions helps us analyze and compare them effectively. Even minor differences in polynomial terms or coefficients can lead to significant changes in the shape and graph of the function.

Function Transformation

Function transformation involves changing the appearance or position of a graph without altering its general form. Common transformations include translations, reflections, stretches, and compressions.

• Translation involves shifting the graph horizontally or vertically. For instance, adding to the input of a function like \(g(x+2)\) results in a horizontal shift.
• If a function is vertically stretched or compressed, it changes the function's steepness.

In the context of our exercise, the function \(g(x+2) = x^2 + 2\) initially appears as a horizontal shift. By re-expressing the function in terms of \(u\) (\(u = x+2\)) and simplifying, it's transformed to \(g(u) = u^2 - 4u + 6\). This demonstrates not just a translation, but also an alteration in the polynomial structure due to additional terms.
Transformations provide insight into how functions are related and help determine equivalence or differences in functional forms.

Equivalent Expressions

Equivalent expressions are different algebraic expressions that have the same value for all values of the variables involved. Identifying equivalent expressions is central to comparing functions.

• A consistent method to check for equivalency is to simplify both expressions as much as possible. They must simplify to the same form for equivalency.
• Factors, terms, and coefficients need to align perfectly.

In the exercise, we compared \(f(x) = x^2\) and \(g(u) = u^2 - 4u + 6\). Since \(g(u)\) has extra terms, it is not equivalent to \(f(x)\). Despite an equal degree in the highest term, the presence of different coefficients and additional terms disqualifies equivalency.
Understanding equivalent expressions aids in recognizing when functions are fundamentally different, even if they might appear similar or undergo specific transformations.