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Chapter 5: Problem 84

Given \(f(x)=x^{2}+7,\) find and simplify. a) \(f(p+1)\) b) \(f(a+h)-f(a)\) c) \(f(a)-f(a-h)\)

Short Answer

Expert verified
a) \( p^2 + 2p + 8 \) b) \( 2ah + h^2 \) c) \( 2ah - h^2 \)

Step by step solution

01

Understand the Function

Given the function: \( f(x) = x^2 + 7 \) We need to find and simplify the given expressions based on this function.
02

Evaluate f(p+1)

To find \( f(p+1) \), substitute \( p+1 \) into \( f(x) \): \[ f(p+1) = (p+1)^2 + 7 \]Expanding the square, we get: \[ (p+1)^2 = p^2 + 2p + 1 \]So, \[ f(p+1) = p^2 + 2p + 1 + 7 = p^2 + 2p + 8 \]
03

Evaluate f(a + h) - f(a)

To find \( f(a + h) - f(a) \), first substitute \( a + h \) and \( a \) into \( f(x) \): \[ f(a + h) = (a + h)^2 + 7 = a^2 + 2ah + h^2 + 7 \]And, \[ f(a) = a^2 + 7 \]Subtract the two expressions to get: \[ f(a + h) - f(a) = (a^2 + 2ah + h^2 + 7) - (a^2 + 7) = 2ah + h^2 \]
04

Evaluate f(a) - f(a - h)

To find \( f(a) - f(a - h) \), first substitute \( a \) and \( a - h \) into \( f(x) \): \[ f(a - h) = (a - h)^2 + 7 = a^2 - 2ah + h^2 + 7 \]And, \[ f(a) = a^2 + 7 \]Subtract the two expressions to get: \[ f(a) - f(a - h) = (a^2 + 7) - (a^2 - 2ah + h^2 + 7) = 2ah - h^2 \]

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Key Concepts

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

Substitution
Substitution is a fundamental technique in mathematics that involves replacing one variable or expression with another. This is particularly useful when working with functions. To substitute, simply replace the variable in the function with the given value or expression.

For example, in the exercise above, we have a function: \(f(x) = x^2 + 7\).
To find \(f(p+1)\), substitute \(p+1\) for x:
\[ f(p+1) = (p+1)^2 + 7 \]
Then, expand and simplify the expression:
\[ (p+1)^2 = p^2 + 2p + 1 \]
So, \(f(p+1) = p^2 + 2p + 8\).

This method helps you evaluate functions at specific points, which is crucial for solving many types of problems.
Simplifying Expressions
Simplifying expressions means to make them easier to work with by combining like terms, reducing fractions, or factoring. It makes complex-looking equations easier to understand.

In the provided exercise, after substitution, the terms are expanded and simplified for clarity. For instance, for \(f(a + h) - f(a)\):
First, substitute and expand:
\[ f(a + h) = (a + h)^2 + 7 = a^2 + 2ah + h^2 + 7 \]
Then, subtract the expressions:
\[ f(a + h) - f(a) = (a^2 + 2ah + h^2 + 7) - (a^2 + 7) = 2ah + h^2 \]
This simplification processes ensures that every term is as straightforward as possible, paving the way for further operations if needed.
Difference of Functions
The difference of functions involves subtracting one function from another. This concept is important for determining how much one function changes relative to another.

In the original exercise, the difference between functions is used in parts b) and c). For example,:

In part b, to find \( f(a + h) - f(a)\), the steps are:
  • Evaluate \(f(a+h)\)
  • Evaluate \(f(a)\)
  • Subtract the two results:
    \[ f(a + h) - f(a) = (a^2 + 2ah + h^2 + 7) - (a^2 + 7) = 2ah + h^2 \]

For part c, follow a similar process to find \(f(a) - f(a - h)\):
  • Evaluate \(f(a-h)\)
  • Evaluate \(f(a)\)
  • Subtract the two results:
    \[ f(a) - f(a - h) = (a^2 + 7) - (a^2 - 2ah + h^2 + 7) = 2ah - h^2 \]

This principle allows you to measure changes between different points, aiding in understanding the behavior of functions.

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