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Chapter 3: Problem 308

In the following exercises, evaluate the function: (@) \(f(2)\) (b) \(f(-1)\) (c) \(f(a) .\) $$ f(x)=3 x+4 $$

Short Answer

Expert verified
f(2) = 10, f(-1) = 1, f(a) = 3a + 4

Step by step solution

01

Evaluate f(2)

Substitute 2 for x in the function: 3(2) + 4 6 + 4 10
02

Evaluate f(-1)

Substitute -1 for x in the function: 3(-1) + 4 -3 + 4 1
03

Evaluate f(a)

Substitute 'a' for x in the function: 3(a) + 4 3a + 4

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Key Concepts

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

substitution method
When evaluating a function, the substitution method can be very helpful. This method involves replacing the variable in the function with a given value.
Let's break it down:
  • Identify the value you need to substitute.
  • Replace the variable in the expression with this value.
  • Perform the arithmetic operations.

For example, if you need to find the value of the function at x = 2, you simply substitute 2 for x in the function.
linear functions
Linear functions represent a straight-line relationship in a graph. The function provided in this exercise, given by the equation
\ f(x) = 3x + 4
is an example of a linear function.
Key characteristics of linear functions include:
  • They always graph to a straight line.
  • The variable `x` has at most power 1.
  • The general form is \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

In our example function, the value of the slope \(m\) is 3, and the y-intercept is 4.
function notation
Function notation is a way of representing functions in a clear and organized manner. When you see
\ f(x)
it means that `f` is the name of the function and `x` is the input variable. This notation helps in understanding and working with functions better.
Here鈥檚 a breakdown:
  • \(f\) denotes the function itself.
  • \(x\) is the input to the function.
  • The output is determined by substituting \(x\) into the function's formula.

For example, in the function \(f(x) = 3x + 4\), if \(x = 2\), we compute \(f(2)\) by substituting 2 into the formula to get \(3(2) + 4\).
simplifying expressions
Simplifying expressions is an important aspect of evaluating functions. It involves performing all the arithmetic operations in the right order to get the final value.
Follow these steps:
  • Substitute the given value into the function.
  • Carry out any multiplications or divisions first.
  • Then, perform any additions or subtractions.

For instance, when evaluating \(f(2)\) for the function \(f(x) = 3x + 4\):
Substituting 2 for x, we get \(3(2) + 4\).
First, compute the product: \(3 \times 2 = 6\).
Then, add 4: \(6 + 4 = 10\).
Hence, \(f(2) = 10\). Simplifying ensures you get the correct answer efficiently.

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Most popular questions from this chapter

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=|x|-1 $$

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