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Chapter 5: Problem 14

Exercise 12 taking \(f(x)=1+2 x\).

Short Answer

Expert verified
For x values substituted into the function \(f(x) = 1 + 2x\), the output is computed by doubling the x value and adding to 1.

Step by step solution

01

Identify the Function

The given function is \(f(x) = 1 + 2x\). The function is linear because there is only one variable \(x\) and the power of the variable is 1.
02

Substitute for x

Given an input for the function, x gets replaced with this input value. For instance, if x = 3, the function becomes \(f(3) = 1 + 2*3\).
03

Compute the Function

Perform the necessary calculations. In this example, \(f(3) = 1 + 6 = 7\). So, when x is 3, \(f(x)\) is 7.

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Key Concepts

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

Function Evaluation
When working with functions, specifically linear functions, understanding function evaluation is crucial. Function evaluation is the process of replacing the variable in a function with a specific input to determine the output. For linear functions like \(f(x) = 1 + 2x\), this means substituting different values for \(x\) and calculating the result.
To evaluate a function:
  • Identify the function you are working with.
  • Select the value for \(x\) you want to evaluate.
  • Replace \(x\) in the function with the chosen value, and solve the resulting expression.
By following these steps, you can assess how the output of the function changes as you input various values of \(x\). This is particularly useful in understanding how functions behave and predicting future outputs.
Substitution Method
The substitution method is a fundamental technique used in evaluating functions, particularly linear ones. It involves replacing the variable in a function with a specific value. For example, in the function \(f(x) = 1 + 2x\), if you want to find \(f(3)\), you would substitute \(3\) for \(x\), resulting in \(f(3) = 1 + 2 \cdot 3\).

The process includes:
  • Identifying the function.
  • Choosing the specific value to substitute for \(x\).
  • Rewriting the equation with the selected value in place of \(x\).
  • Calculating the output by performing the arithmetic operations.
This method is widely used because it's straightforward and simplifies the process of evaluating functions with various inputs. Understanding this method allows you to handle more complex algebraic expressions with confidence.
Algebraic Expressions
Algebraic expressions are key components in linear functions. They are mathematical phrases that can contain numbers, variables, and operators like addition or multiplication. In the context of our function \(f(x) = 1 + 2x\), the expression "\(1 + 2x\)" is an algebraic expression.

These expressions represent relationships, and when evaluated with a substitution, they reveal the function's output.
  • An algebraic expression can be evaluated by substituting variables with specific values.
  • Perform arithmetic operations to simplify the expression and arrive at a result.
This combination of numbers and symbols forms the basis for solving equations and evaluating functions. By understanding algebraic expressions, you can effectively apply the substitution method and evaluate functions to predict outcomes based on inputs.

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Most popular questions from this chapter

Use a CAS to find \(f\) from the information given. $$f^{\prime}(x)=3 \sin x+2 \cos x ; \quad f(0)=0, f^{\prime}(0)=0$$

Suppose that \(f\) is continuous on \([a, b]\) and \(\int_{a}^{b} f(x) d x=0\) Prove that there is at least one number \(c\) in \((a, b)\) for which \(f(c)=0\).

Use a CAS to find \(f\) from the information given. $$f^{\prime}(x)=5-3 x+x^{2} ; \quad f(0)=-3, f^{\prime}(0)=4$$

Calculate. $$\int \sin ^{2} \pi x \cos \pi x d x$$

Exercise 38 taking \(f(x)=\sin x\) with \(x \in[0, \pi]\).
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