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Chapter 9: Problem 7

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1\) \(f(-3)\)

Short Answer

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Step by step solution

01

- Identify the function f(x)

The function given is \(f(x) = -3x + 4\).
02

- Substitute the given value into f(x)

Substitute \(x = -3\) into the function \(f(x)\).
03

- Perform the calculation

Calculate \(f(-3) = -3(-3) + 4\).
04

- Simplify the expression

Simplify \(-3(-3) + 4\) to obtain \(9 + 4\).
05

- Write the final result

The final result is \(9 + 4 = 13\).

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Key Concepts

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

Function Notation
Function notation is a compact way to represent functions, where we write functions as f(x), g(x), etc., instead of traditional y. Here, f and g are the names of the respective functions, and x is the variable.

For example, in the problem given, we have:
  • f(x) = -3x + 4
  • g(x) = -x^2 + 4x + 1
Function notation helps to clearly indicate which function is being worked on and makes it easy to substitute values into the functions.

You can think of f(x) as 'f of x' which tells us that x is the input, and the expression on the right side of the equation is the rule we follow to get the output. When the function notation says f(-3), it means we should replace every x in the expression with -3.
Substitution Method
The substitution method involves replacing the variable in the function notation with a specific number. This process allows us to evaluate the function at that particular value.

Let's walk through the example step by step. We have the function f(x) = -3x + 4 and need to find f(-3). Here鈥檚 how we substitute the method:
  • Step 1: Identify the function - Here, f(x) = -3x + 4.
  • Step 2: Substitute the given value - We will replace x with -3.
  • Step 3: f(-3) = -3(-3) + 4.
  • Step 4: Perform the calculation -3(-3) + 4.
Thus, when we substitute x = -3 into the function f(x), we calculate -3(-3) + 4 to get 9 + 4, which simplifies to 13.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and the operations performed on them (e.g., addition, subtraction). In our context, we have the expression -3x + 4.

When solving these expressions, we follow certain rules:
  • Perform multiplication and division before addition and subtraction.
  • Follow the order of operations (PEMDAS/BODMAS).
In the given example, substituting x with -3, we get -3(-3) + 4:
  • First, multiply: -3 * -3 = 9.
  • Then, add: 9 + 4 = 13.
Understanding how to manipulate algebraic expressions correctly is crucial for evaluating functions. It allows us to accurately simplify and solve the expressions, arriving at the correct result.

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Most popular questions from this chapter

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{h}{g}\right)(x) $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{f}{g}\right)(x) $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ h)(0) $$

In a sale room at a clothing store, every item is on sale for half its original price, plus \(\$ 1 .\) (a) Write a function \(g\) that finds half of \(x\). (b) Write a function \(f\) that adds 1 to \(x\). (c) Write and simplify the function \((f \circ g)(x)\). (d) Use the function from part (c) to find the sale price of a shirt that has an original price of \(\$ 60\).

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (h \circ g)(x) $$
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