Problem 25 - Algebra Form and Function

Question

Find a possible formula for \(f\) given that $$ \begin{aligned} f\left(x^{2}\right) &=2 x^{4}+1 \\ f(2 x) &=8 x^{2}+1 \\ f(x+1) &=2 x^{2}+4 x+3 \end{aligned} $$

Step-by-Step Solution

Answer

Question: Determine the formula for the function \(f(x)\), if the following expressions hold: \(f\left(x^{2}\right) = 2x^{4} + 1\), \(f(2x) = 8x^{2} + 1\), and \(f(x+1) = 2x^{2} + 4x + 3\). Answer: The formula for the function \(f(x)\) is \(f(x) = 2x^{2} + 1\).

Step 1: Observing the given expressions

First, let's write down the given expressions. $$ \begin{aligned} f\left(x^{2}\right) &= 2x^{4} + 1 \\\ f(2x) &= 8x^{2} + 1 \\\ f(x+1) &= 2x^{2} + 4x + 3 \end{aligned} $$

Step 2: Transform the second expression \(f(2x)\)

To find a pattern, we can first manipulate the second expression \(f(2x) = 8x^{2} + 1\) by substituting \(x^{2}\) in place of \(x\), i.e., we will replace \(x\) with the square root of \(x^{2}\): $$ f(2\sqrt{x^{2}}) = 8x^{2} + 1 $$

Step 3: Compare the first and modified second expressions

Next, let's compare the first expression, \(f\left(x^{2}\right)\), and the modified second expression, \(f(2\sqrt{x^{2}})\): $$ \begin{aligned} f\left(x^{2}\right) &= 2x^{4} + 1 \\ f(2\sqrt{x^{2}}) &= 8x^{2} + 1 \end{aligned} $$ We notice that both expressions have a similar structure, except that the second expression has a "^{4}" replaced with a "^{2}". So, we can express the function as: $$ f(kx^n) = 2k^{\frac{4}{n}}x^{4} + 1 $$ where \(k\) and \(n\) are constants.

Step 4: Check the third expression

Now, let's try to fit the third expression, \(f(x+1)\), to the function we derived: $$ f(x+1) = 2(x+1)^{2} + 4(x+1) + 3 $$ We see that this expression doesn't fit our derived pattern, so we need to come up with a new approach.

Step 5: Finding a general pattern

Let's consider a quadratic function for \(f(x)\): $$ f(x) = ax^{2} + bx + c $$ Our goal is to find the values of \(a\), \(b\), and \(c\) that satisfy the given expressions.

Step 6: Apply the general pattern to the expressions

We need to re-write the given expressions using the general pattern \(f(x) = ax^{2} + bx + c\) by substituting the given inputs in the function: $$ \begin{aligned} f\left(x^{2}\right) = ax^{4} + bx^{2} + c &= 2x^{4} + 1 \\ f(2x) = a(2x)^{2} + b(2x) + c &= 8x^{2} + 1 \\ f(x+1) = a(x+1)^{2} + b(x+1) + c &= 2x^{2} + 4x + 3 \end{aligned} $$

Step 7: Solve the system of equations

Now, we have three equations with three unknowns: $$ \begin{aligned} ax^{4} + bx^{2} + c &= 2x^{4} + 1 \\ 4ax^{2} + 2bx + c &= 8x^{2} + 1 \\ a(x+1)^{2} + b(x+1) + c &= 2x^{2} + 4x + 3 \end{aligned} $$ Comparing the coefficients of the powers of \(x\), we have: $$ \begin{cases} a = 2 \\ 4a + 2b = 8 \\ a + b + c = 3 \end{cases} $$ Solving this system of equations, we get \(a = 2\), \(b = 0\), and \(c = 1\).

Step 8: Express the function \(f(x)\) using the obtained values

With the obtained values of \(a\), \(b\), and \(c\), we can now express the function \(f(x)\): $$ f(x) = ax^{2} + bx + c = 2x^{2} + 0x + 1 = 2x^{2} + 1 $$ Thus, a possible formula for \(f(x)\) is: $$ f(x) = 2x^{2} + 1 $$

Key Concepts

Function Notation

Function notation is a convenient way to express relationships between quantities. When you see a notation like \( f(x) \), it represents a function named \( f \) with an input \( x \). This notation helps us define a rule or formula where each input is assigned to exactly one output.
For instance, in the context of quadratic functions like \( f(x) = 2x^2 + 1 \), \( x \) is the variable we can replace with any number to find the corresponding function value.
This makes it very visual and easy to track transformations and calculations.

• The input \( x \) is placed in the parentheses.
• The rule or formula is given on the right side.

Function notation is fundamental because it clearly states the role of the input and output in equations, helping us to maintain a clear understanding of how different mathematical expressions interact and are manipulated.

Algebraic Expressions

Algebraic expressions play a critical role in mathematics as they consist of numbers, variables, and the operations that apply to them like addition, subtraction, multiplication, and division.
Expressions like \( 2x^2 + 1 \) show how quantities relate to each other using formulas and make visible the patterns in values that we see in function equations.
You can think of an algebraic expression as a phrase in a sentence, where the numbers and variables provide meaning, and the operations link these parts together.
In dealing with quadratic functions:

• The expression \( 2x^2 \) shows how the squared variable \( x \) is scaled by the coefficient 2.
• The constant 1 adjusts the result, giving the final output of the function for a given \( x \).

Understanding how to manipulate these expressions allows us to solve equations and derive new equations helpful in optimizing various real-life situations.

Systems of Equations

Systems of equations occur when we have multiple equations that all use the same variables. To solve them, we need to find a set of values that satisfy all equations simultaneously.
In our quadratic function example, we had three different expressions given:

• \( f(x^2) = 2x^4 + 1 \)
• \( f(2x) = 8x^2 + 1 \)
• \( f(x+1) = 2x^2 + 4x + 3 \)

Given these, we derived a system of equations to find the constants \( a \), \( b \), and \( c \) in our general quadratic equation format of \( f(x) = ax^2 + bx + c \). Solving this system involves:

• Substituting each expression into the general form.
• Finding the values that satisfy all equations.
• Ensuring the solution is consistent across all provided equations.

Systems of equations are widely used in different fields such as science, engineering, and economics to model and solve complex problems involving multiple conditions.