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Chapter 5: Problem 11

Let \(f(x)=-3 x+1\) and \(g(x)=x^{2}+2 .\) Find each of the following. $$ f(-1) \cdot g(-1) $$

Short Answer

Expert verified
12

Step by step solution

01

- Evaluate f(-1)

First, evaluate the function f at x = -1. Substitute -1 into the expression for f: \( f(x) = -3x + 1 \) Thus, \( f(-1) = -3(-1) + 1 = 3 + 1 = 4 \)
02

- Evaluate g(-1)

Next, evaluate the function g at x = -1. Substitute -1 into the expression for g: \( g(x) = x^2 + 2 \) Thus, \( g(-1) = (-1)^2 + 2 = 1 + 2 = 3 \)
03

- Multiply the Results

Now, multiply the results from Step 1 and Step 2 together: \( f(-1) \times g(-1) = 4 \times 3 = 12 \)

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Key Concepts

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

evaluating functions
Evaluating functions means finding the value of a function for a specific input. In the given exercise, we have two functions:
  • Function f: \( f(x) = -3x + 1 \)
  • Function g: \( g(x) = x^{2} + 2 \)
To evaluate these functions at a specific input, follow these steps:
  • First, substitute the given input into the function.
  • Next, perform the arithmetic operations.
For example, to evaluate f at -1, we substitute -1 into the expression \( f(x) \to f(-1) = -3(-1) + 1 = 3 + 1 = 4 \). Similarly, to evaluate g at -1, we substitute -1 into the expression \( g(x) \to g(-1) = (-1)^2 + 2 = 1 + 2 = 3 \).
multiplying functions
Multiplying functions involves finding the product of the values of each function at a given input. After evaluating both functions at the specified input, simply multiply the results together. In the present example, we first find the values of \( f(-1) = 4 \) and \( g(-1) = 3 \). Then, to find the product, we multiply these values:
  • \( f(-1) \times g(-1) = 4 \times 3 = 12 \)
This final step gives us the result of the function multiplication at the chosen input.
basic algebraic operations
Basic algebraic operations are essential skills in algebra that include adding, subtracting, multiplying, and dividing numbers and expressions. In the exercise, we specifically look at multiplication and substitution.
  • Substitution: Replace the variable with a given number. For example, substituting -1 for x in \( f(x) = -3x + 1 \) gives \( -3(-1) + 1 \).
  • Multiplication: Multiply results from evaluating functions. Our example shows the multiplication of the results from two function evaluations: **4** from f(-1) and **3** from g(-1).
These operations form the foundation of algebra and are used to solve more complex problems systematically.

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