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Chapter 0: Problem 10

Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=3 x-1\) \(\frac{f(x)-f(1)}{x-1}\)

Short Answer

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Step by step solution

01

Evaluate function at \(x = 1\)

The function \(f(x) = 3x - 1\). We are required to find the value of the function at \(x = 1\). Therefore, we replace \(x\) with 1 in our function. Hence, we have \(f(1) = 3*1 - 1 = 2\).
02

Write down the expression

Next we replace \(f(x)\) and \(f(1)\) with their respective values in the expression \(\frac{f(x)-f(1)}{x-1}\). Which results in \(\frac{3x - 1 - 2}{x - 1} = \frac{3x - 3}{x - 1}\).
03

Simplify

Here, both the numerator and denominator can be simplified. This will give us 3. Hence, our expression simplifies to 3. Thus, the function is continuous at \(x = 1\)

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Key Concepts

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

Functions
A function in calculus is a relationship between two sets. Each input from the first set (called the domain) maps to one output in the second set (called the range). In simple terms, a function tells us how one quantity depends on another. For example, the function given in our problem, \(f(x) = 3x - 1\), implies that for every value of \(x\) (our input), we calculate the corresponding output using the expression on the right-hand side. Functions are essential in various areas of mathematics and are used to model real-world scenarios. Knowing how to evaluate functions at specific points鈥攁s shown when finding \(f(1) = 2\)鈥攊s a fundamental skill that underpins much of calculus.
Limits
Limits help us understand the behavior of functions as they approach specific values. In calculus, a limit examines what happens to a function as the input approaches a particular point. For the expression \(\frac{f(x)-f(1)}{x-1}\), the limit as \(x\) approaches 1 tells us how the function behaves near that point, even if \(x = 1\) itself might create an undefined or indeterminate form. Evaluating this limit requires simplifying the expression, reflecting the function's behavior close to that point.
  • Limits are useful when we want to analyze points where functions aren鈥檛 directly defined.
  • They also form the basis for defining derivatives and continuity.
Understanding limits enables us to deal with expressions that initially seem indeterminate, like \(\frac{0}{0}\).
Continuity
Continuity of a function at a point means there's no interruption in its graph at that location. A continuous function travels from one point to another without breaks or gaps. To determine continuity at \(x=1\), we need to ensure three conditions:
  • The function \(f(x)\) must be defined at \(x=1\).
  • The limit of \(f(x)\) as \(x\) approaches 1 must exist.
  • The value of the function at that point, \(f(1)\), must equal the limit found.
Since \(\frac{f(x)-f(1)}{x-1}\) simplifies to 3 (a constant), the limit as \(x\) approaches 1 is 3. Thus, \(f(x)\) is continuous at this point, making it smooth and predictable without any sudden jumps.
Algebraic Simplification
Algebraic simplification is the process of rewriting an expression into its simplest form. This makes it easier to understand and work with. In our problem, after substituting \(f(x)\) and \(f(1)\) into the expression \(\frac{f(x)-f(1)}{x-1}\), we rearrange and simplify:
  • The numerator \(3x-3\) gets simplified by factoring out a constant: \(3(x-1)\).
  • The denominator \(x-1\) remains the same.
Since the \(x-1\) in the numerator and the denominator cancel out, we are left with 3. Simplifying such expressions is crucial for finding direct answers, recognizing patterns, and solving complex problems easily.

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Most popular questions from this chapter

Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=\sin x\) (a) \(f(\pi)\) (b) \(f(5 \pi / 4)\) (c) \(f(2 \pi / 3)\)

Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$g(x)=\frac{4}{x}$$

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Find the composite functions \((f \circ g)\) and \((g \circ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=\frac{3}{x}\) \(g(x)=x^{2}-1\)

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