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Chapter 9: Problem 40

$$ \frac{1}{x-3} ; a=1 $$

Short Answer

Expert verified
The value of the function at a = 1 is -1/2.

Step by step solution

01

Understand the Given Function

We are given the function \( f(x) = \frac{1}{x-3} \) and we need to evaluate or understand this function when \( a = 1 \). This means substituting \( x = 1 \) into the function.
02

Substitute the Given Value

Substitute \( x = 1 \) into the function: \( f(x) = \frac{1}{x - 3} \). This becomes \( f(1) = \frac{1}{1 - 3} \).
03

Simplify the Expression

Simplify the expression \( f(1) = \frac{1}{1 - 3} \). Perform the subtraction in the denominator: \( 1 - 3 = -2 \). Therefore, \( f(1) = \frac{1}{-2} \).
04

Write the Final Answer

The value of the function \( f(x) \) at \( a = 1 \) is \( f(1) = -\frac{1}{2} \). This gives us the function's value at this specific point.

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Key Concepts

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

Evaluating functions at a point
When we talk about evaluating functions at a point, we are essentially interested in finding the output value of a function for a particular input. A function, such as \( f(x) = \frac{1}{x-3} \), relies on its input \( x \) to produce an output. By specifying \( a = 1 \), you are instructed to calculate what the function yields when \( x = 1 \).
  • First, replace \( x \) with the given number, which is \( a = 1 \). This transforms the function to \( f(1) = \frac{1}{1-3} \).
  • The main goal is to compute the result of this transformation to find the particular value at point \( a \).
Computing the value at this specific input allows you to understand how the function behaves at that point. It's like checking the function's response to a specific signal.
Simplifying expressions
Simplifying expressions is like cleaning up math to make it more understandable. It means performing all possible arithmetic and reducing fractions when you substitute a number into a function. In this case, after inserting \( x = 1 \) into the function \( f(x) = \frac{1}{x-3} \), the expression becomes \( f(1) = \frac{1}{1-3} \).
  • Start with solving the subtraction in the denominator: \( 1 - 3 \).
  • Calculate \( 1 - 3 = -2 \).
  • The fraction \( \frac{1}{-2} \) simplifies to the consistent and final result of \( -\frac{1}{2} \).
Through steps like these, you make the function's output easily interpretable and concise.
Substitution in functions
Substitution in functions is a method where you replace the variable in a function with a particular value. This technique allows you to pinpoint exactly what the function equals at specific inputs. In this exercise, the function is \( f(x) = \frac{1}{x-3} \), and you're told to substitute \( x = 1 \).
  • Switch out the \( x \) with the value you're given, which turns the function into \( f(1) = \frac{1}{1-3} \).
  • The substitution is a direct swap inside the expression, facilitating the evaluation of the function at that point.
Doing this correctly paves the way for you to process and understand the essence of the function鈥檚 behavior at that specific input.

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Most popular questions from this chapter

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{(x-2)^{n}}{n} $$

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ 1+(x+2)+\frac{(x+2)^{2}}{2 !}+\frac{(x+2)^{3}}{3 !}+\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\left(\frac{1}{4}\right)^{n}+3^{n / 2}\)

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\frac{x^{8}}{8 !}-\frac{x^{10}}{10 !}+\cdots $$

In Problems 1-20, an explicit formula for \(a_{n}\) is given. Write the first five terms of \(\left\\{a_{n}\right\\}\), determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\) \(a_{n}=\frac{n^{100}}{e^{n}}\)
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