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Chapter 6: Problem 53

Simplify the given expressions. $$\text { If } f(x)=x-\frac{2}{x}, \text { find } f(a+1)$$

Short Answer

Expert verified
\( f(a+1) = a + 1 - \frac{2}{a+1} \)

Step by step solution

01

Substitute into the function

Identify the function given as \( f(x) = x - \frac{2}{x} \). To find \( f(a+1) \), substitute \( a+1 \) in place of \( x \) in the function. This will give \( f(a+1) = (a+1) - \frac{2}{a+1} \).
02

Simplify the expression

Rewrite \( f(a+1) = a + 1 - \frac{2}{a+1} \). This consists of two parts: \( a+1 \) and \(-\frac{2}{a+1} \). There is no further simplification needed because the terms cannot be combined without a common denominator, so the simplified expression is: \[ f(a+1) = a + 1 - \frac{2}{a+1}. \]

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Key Concepts

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

Substitution
When dealing with functions, substitution is a fundamental technique. Imagine that a function is like a machine. You put something in it, and it gives you an output. In the context of the exercise, we have a specific function: \( f(x) = x - \frac{2}{x} \). To find a particular value, we substitute the value of \( x \) with what we are interested in. For this exercise, we wanted to find \( f(a+1) \). This means we take the expression \( a+1 \) and put it wherever there is an \( x \) in the function.
  • Think of substitution as a replacement.
  • Substitute \( a+1 \) in place of every \( x \) in the original function.
  • This gives us the expression \( f(a+1) = (a+1) - \frac{2}{a+1} \).
Mastering substitution is crucial because it allows us to work with functions in practical ways, answering specific questions or finding particular values.
Expressions
An expression in mathematics is a combination of numbers, symbols, and operators that collectively show a value. In our exercise, \( f(a+1) = a + 1 - \frac{2}{a+1} \) is an expression. Expressions can vary in complexity and setup, but understanding their components is pivotal.
  • Basic expressions might just involve numbers and one operation, such as addition \( 3 + 5 \).
  • More complex expressions include variables, like our expression where both \( a \) and 1 are variables within \( a+1 \).
  • The expression combines operations of addition and subtraction with elements like fractions.
Recognizing and understanding expressions helps in simplifying them and finding their values under different conditions.
Simplification
Simplification is a process where we make an expression easier to work with. In our exercise, we were given the expression \( a+1 - \frac{2}{a+1} \) after substituting \( a+1 \) into the function. Simplification involves steps to make calculations straightforward without changing the expression's original value.
  • Simplification often involves combining like terms, which are terms in an expression that have the same variable part.
  • If fractions are involved, simplification may involve finding common denominators to combine terms.
  • In our specific problem, the expression is simplest in its given form as \( a + 1 - \frac{2}{a+1} \) since there's no straightforward way to combine terms further without altering its fundamental components.
Thus, a simplified expression is just as valuable and often easier to use in further calculations.
Denominator
The denominator is the bottom part of a fraction. It's critical in expressions as it divides the numerator, giving fractional expressions a unique form. In our exercise, the fraction \( \frac{2}{a+1} \) is present.
  • The denominator \( a+1 \) means that whatever it divides, makes it a part of a regional piece of a whole, essentially showing how large or small it is in comparison.
  • The denominator determines the division's scale in a fraction, affecting how terms can be simplified.
  • A larger denominator means smaller slices, while smaller denominators indicate larger slices when visualizing fractional amounts.
Understanding denominators in expressions is crucial for simplification processes, especially when working with complex fractions.

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