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Chapter 5: Problem 90

Evaluate \((a) f(-1)\) and \((b) f(2)\) for each function. $$ f(x)=3 x+1 $$

Short Answer

Expert verified
For \[f(x) = 3x + 1\], \[f(-1) = -2\] and \[f(2) = 7\].

Step by step solution

01

Title - Identify the Function

The given function is \[f(x) = 3x + 1\]Let's evaluate it for specific values of x.
02

Title - Evaluate f(-1)

To find \[f(-1)\], substitute -1 for x in the function:\[f(-1) = 3(-1) + 1\]Simplify:\[f(-1) = -3 + 1 = -2\]Therefore, \[f(-1) = -2\]
03

Title- Evaluate f(2)

To find \[f(2)\], substitute 2 for x in the function:\[f(2) = 3(2) + 1\]Simplify:\[f(2) = 6 + 1 = 7\]Therefore, \[f(2) = 7\]

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Key Concepts

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

substitution method
The substitution method is a valuable technique used to evaluate functions. It involves replacing the variable in a function with a specific value. This method allows us to determine the function's output for different inputs.

For example, given the function \[ f(x) = 3x + 1 \], we can use substitution to find the values of \[ f(-1) \] and \[ f(2) \]:

To evaluate \[ f(-1) \]:
  • Substitute -1 for x in the function.
  • Calculate \[ f(-1) = 3(-1) + 1 = -3 + 1 = -2 \]
  • Thus, \[ f(-1) = -2 \]
To evaluate \[ f(2) \]:
  • Substitute 2 for x in the function.
  • Calculate \[ f(2) = 3(2) + 1 = 6 + 1 = 7 \]
  • Thus, \[ f(2) = 7 \]
By using the substitution method, we can easily find the specific outputs of a function by plugging in known values.
linear functions
Linear functions are one of the simplest types of functions. They can be represented by a straight line when graphed. A linear function is usually in the form of \[ f(x) = mx + b \], where:
  • \(\text{m}\) is the slope, which indicates the steepness and direction of the line.
  • \(\text{b}\) is the y-intercept, which is the point where the line crosses the y-axis.
In our example, the function is \[ f(x) = 3x + 1 \] where:
  • The slope (m) is 3, meaning the line rises 3 units for every unit it moves to the right.
  • The y-intercept (b) is 1, so the line crosses the y-axis at the point (0, 1).
Understanding linear functions is crucial because they appear frequently in algebra and real-life situations. They make it easier to predict and analyze relationships between variables.
function evaluation
Function evaluation involves finding the output of a function for a given input. This process helps understand the behavior of functions and is critical in many areas of math and science.

To evaluate a function like \[ f(x) = 3x + 1 \], follow these steps:
  • Identify the function you're working with.
  • Substitute the given input value into the function.
  • Perform the arithmetic operations to find the output.
For example, to evaluate f(-1):
  • Substitute -1 into the function: \[ f(-1) = 3(-1) + 1 \]
  • Calculate \[ f(-1) = -3 + 1 = -2 \]
Similarly, to evaluate f(2):
  • Substitute 2 into the function: \[ f(2) = 3(2) + 1 \]
  • Calculate \[ f(2) = 6 + 1 = 7 \]
By practicing function evaluation, students can better understand how functions work and how to apply them in different contexts.

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