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Chapter 14: Problem 84

If \(f(x)=x^{2}+2 x+3,\) find \(f(a+1)\) (Section \(8.1,\) Example 3 )

Short Answer

Expert verified
So, \(f(a+1) = a^{2} + 4a + 6\).

Step by step solution

01

Substituting \(x\) with \(a+1\)

Replace \(x\) in the function \(f(x)=x^{2}+2 x+3\) with \(a+1\) to get \(f(a+1) = (a+1)^{2} + 2*(a+1) + 3\)
02

Expanding the Formula

Expand the square and the multiplication to get \(f(a+1) = a^{2} + 2a + 1 + 2a + 2 + 3\)
03

Simplifying the Expression

Combine like terms to simplify the expression to get \(f(a+1) = a^{2} + 4a + 6\)

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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 can be used to describe a variety of real-world situations. They consist of variables raised to different powers and multiplied by coefficients. For example, the function \( f(x) = x^2 + 2x + 3 \) is a polynomial function where \( x^2 \) is the quadratic term, \( 2x \) is the linear term, and \( 3 \) is the constant term.
Polynomials are powerful because they are smooth and continuous functions, which make them easy to manipulate mathematically.
  • The degree of a polynomial is determined by the highest power of the variable in the expression. In our example, the degree is 2, making it a quadratic polynomial.
  • Polynomials can be added, subtracted, multiplied, and even divided (though division by polynomials can get into more complex areas like polynomial long division).
Understanding polynomial functions is important for calculus and many fields of science and engineering.
Substitution Method
The substitution method is a technique used to evaluate functions at specific points or to transform the expression inside a function. In the example given, we are asked to find \( f(a+1) \) for the polynomial \( f(x) = x^2 + 2x + 3 \).
To do this, everywhere the variable \( x \) appears in the polynomial, we will replace it with \( a+1 \). This gives us:
  • Start with the expression \( f(a+1) = (a+1)^2 + 2(a+1) + 3 \).
  • Pay attention to the entire expression being substituted, including brackets which maintain the integrity of operations when expanded later.
Substitution is particularly helpful when you need to evaluate a polynomial function at points other than the typical \( x \)-values (such as integers).
Simplifying Expressions
After performing a substitution in a polynomial, the next step is often to simplify the resulting expression. Simplification makes the expression easier to interpret and analyze.
The example starts with the substituted form \( f(a+1) = (a+1)^2 + 2(a+1) + 3 \), which means you need to:
  • First, expand each term: \( (a+1)^2 = a^2 + 2a + 1 \).
  • Next, handle the multiplication: \( 2(a+1) = 2a + 2 \).
Once all terms are expanded, combine like terms, which means adding together terms that have the same variables raised to the same power:
  • Combine \( a^2 \), \( 2a + 2a \), and \( 1 + 2 + 3 \).
The simplified form \( f(a+1) = a^2 + 4a + 6 \) is now much easier to understand and work with. Simplifying expressions is a critical skill that aids in solving equations and understanding mathematical models.

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Most popular questions from this chapter

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x-1)^{3}$$

Expand and write the answer as a single logarithm with a coefficient of 1. $$\sum_{i=1}^{4} \log (2 i)$$

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$5,7,9,11, \dots$$

Explain how to find the general term of a geometric sequence.

Will help you prepare for the material covered in the next section. Consider the sequence \(8,3,-2,-7,-12, \ldots .\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?
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