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Chapter 0: Problem 17

If \(f(x)=x^{2}-2 x\), find \(f(a+1)\) and \(f(a+2)\).

Short Answer

Expert verified
f(a+1) = a^2 - 1 and f(a+2) = a^2 + 2a

Step by step solution

01

- Understand the Function

The given function is \( f(x) = x^{2} - 2x \). This means for any input value of \( x \), you substitute \( x \) into \( x^{2} - 2x \) to find the output.
02

- Finding \( f(a+1) \)

To find \( f(a+1) \), substitute \( a+1 \) for \( x \) in the given function: \[ f(a+1) = (a+1)^2 - 2(a+1) \].
03

- Simplify \( f(a+1) \)

Expand and simplify the expression: \[ (a+1)^2 = a^2 + 2a + 1 \]. Now, substitute back into the function and simplify: \[ f(a+1) = a^2 + 2a + 1 - 2a - 2 = a^2 - 1 \]. Thus, \( f(a+1) = a^2 - 1 \).
04

- Finding \( f(a+2) \)

To find \( f(a+2) \), substitute \( a+2 \) for \( x \) in the function: \[ f(a+2) = (a+2)^2 - 2(a+2) \].
05

- Simplify \( f(a+2) \)

Expand and simplify the expression: \[ (a+2)^2 = a^2 + 4a + 4 \]. Now, substitute back into the function and simplify: \[ f(a+2) = a^2 + 4a + 4 - 2a - 4 = a^2 + 2a \]. Thus, \( f(a+2) = a^2 + 2a \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polynomial functions
Polynomial functions are mathematical expressions that involve sums of powers of variables with coefficients. They are written in the form \ ax^{n} + bx^{n-1} + ... + zx^{0} \ where \( a, b, ... , z \) are constants and \( x \) is the variable. For example, in the exercise, the function \( f(x) = x^{2} - 2x \) is a polynomial. The highest power of the variable \( x \) is called the degree of the polynomial. Here, since the highest power of \( x \) is 2, the polynomial is of degree 2.
function substitution
Function substitution is a technique used to replace the variable of a function with another expression. This is essential for evaluating functions for specific arguments.
Let's consider the function \( f(x) = x^{2} - 2x \). If we need to find \( f(a+1) \), we substitute \( a+1 \) for \( x \), giving us: \ f(a+1) = (a+1)^{2} - 2(a+1) \.
This turns the process of function evaluation into one of direct substitution, which is then simplified to get the result. This technique is powerful and can be applied to various functions and expressions.
simplifying expressions
Simplifying expressions involves reducing them to their most basic form for easier evaluation and understanding. Here's a step-by-step approach:
  • Expand all products: For \( f(a+1) \), expand \( (a+1)^{2} \) to get \( a^{2} + 2a + 1 \).
  • Combine like terms: Substitute back into the function and combine like terms. For instance, \( f(a+1) = a^{2} + 2a + 1 - 2a - 2 \) simplifies to \( a^{2} - 1 \).
  • Double-check the result: Make sure there are no further terms that can be combined or simplified. This ensures our evaluation is accurate. By doing this, we simplify the function to create a more straightforward problem to solve.

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Most popular questions from this chapter

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