Problem 88 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } f(x)=x^{2}-3 x+2, \text { find } f(f(x)) $$

Step-by-Step Solution

Answer

The short answer is \(f(f(x)) = x^4 - 6x^3 + 9x^2 - 3x\).

Step 1: Write down the given function and the desired function composition

We are given the function \(f(x) = x^2 - 3x + 2\) and we need to find \(f(f(x))\).

Step 2: Substitute the given function in the place of x

To find \(f(f(x))\), we will substitute the given function \(x^2 - 3x + 2\) in place of x: \[f(f(x)) = f(x^2 - 3x + 2)\]

Step 3: Substitute the expression and simplify

Now, we will substitute the definition of \(f(x)\) into the expression and then simplify: \[f(f(x)) = (x^2 - 3x + 2)^2 - 3(x^2 - 3x + 2) + 2\]

Step 4: Expand the expression and simplify further

Expand and simplify the expression as follows: \begin{align*} f(f(x)) &= (x^2 - 3x + 2)^2 - 3(x^2 - 3x + 2) + 2 \\ &= (x^4 - 6x^3 + 12x^2 - 12x + 4) - (3x^2 - 9x + 6) + 2 \\ &= x^4 - 6x^3 + 12x^2 - 12x + 4 - 3x^2 + 9x - 6 + 2 \\ &= x^4 - 6x^3 + 9x^2 - 3x \\ \end{align*} Therefore, the desired function composition is \(f(f(x)) = x^4 - 6x^3 + 9x^2 - 3x\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions made up of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. These functions can have different degrees depending on their highest power of the variable. For example, in the given function, \(f(x) = x^2 - 3x + 2\), it is a quadratic polynomial where the highest exponent is 2.

Polynomials are generally classified by their degree:

• Linear polynomial: Degree 1 (e.g., \(ax + b\))
• Quadratic polynomial: Degree 2 (e.g., \(ax^2 + bx + c\))
• Cubic polynomial: Degree 3 (e.g., \(ax^3 + bx^2 + cx + d\))

The degree of a polynomial gives us important information about the number of roots or intercepts and the shape of its graph.

Function composition with polynomials, like finding \(f(f(x))\), involves combining and simplifying these polynomial expressions, resulting in new polynomials of higher degrees. Each mathematical operation follows the basic algebraic rules to maintain the proper structure of these functions.

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest degree of the variable is 2. This gives them the standard form \(ax^2 + bx + c\). In the provided function, \(f(x) = x^2 - 3x + 2\), \(a = 1\), \(b = -3\), and \(c = 2\).

Quadratic functions graph into a parabola. The orientation of the parabola (opening upwards or downwards) is determined by the sign of the coefficient \(a\). A positive \(a\) means the parabola opens upwards, while a negative \(a\) opens it downwards.

Key features of quadratics include:

• Vertex: The highest or lowest point of the parabola.
• Axis of symmetry: A vertical line that passes through the vertex and divides the parabola into symmetrical halves.
• Roots (or zeros): Points where the parabola crosses the x-axis, which are solutions to the quadratic equation.

Composing quadratic functions, as in finding \(f(f(x))\), involves substituting and expanding expressions to analyze and create more complex functions, maintaining an understanding of how these parabolas behave graphically.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill needed to solve problems involving functions and equations. It involves rearranging equations, expanding expressions, and simplifying polynomial terms, just like in the process of finding \(f(f(x))\) from the exercise.

Let's break down a basic approach to algebraic manipulation in this context:

• Substitution: Start by replacing the variable in one function with another expression. For \(f(f(x))\), substitute \(f(x)\) into itself.
• Expansion: Apply distributive property to remove brackets, multiplying each term as required. For instance, expanding \((x^2 - 3x + 2)^2\) results in a new polynomial with degree 4.
• Simplification: Combine like terms to simplify the expression. This involves carefully aligning terms of similar powers and reducing to the simplest form.

These skills are essential as they unlock the ability to delve deeper into more complex algebraic problems. Mastering them arms students with tools to handle both straightforward and intricate mathematical tasks, ensuring clarity in manipulating expressions and solving equations.