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Magazine Chapter 2: Problem 2
Find (a) \((f+g)(3)\) (b) \((f-g)(3)\) (c) \((f g)(3)\) (d) \((f / g)(3)\) $$f(x)=-x^{2}, \quad g(x)=2 x-1$$
Short Answer
Expert verified
a) -4, b) -14, c) -45, d) -1.8
Step by step solution
01
Understand the Functions
We have two functions: \(f(x) = -x^2\) and \(g(x) = 2x - 1\). We need to evaluate these functions at \(x = 3\), since we'll be combining them in several ways.
02
Evaluate Individual Functions at x = 3
First, calculate \(f(3)\) and \(g(3)\). For \(f(3)\):\[f(3) = -(3)^2 = -9\]For \(g(3)\):\[g(3) = 2(3) - 1 = 6 - 1 = 5\]
03
Calculate \((f+g)(3)\)
\((f+g)(3)\) is the sum of \(f(3)\) and \(g(3)\). \[(f+g)(3) = f(3) + g(3) = -9 + 5 = -4\]
04
Calculate \((f-g)(3)\)
\((f-g)(3)\) is the difference between \(f(3)\) and \(g(3)\). \[(f-g)(3) = f(3) - g(3) = -9 - 5 = -14\]
05
Calculate \((f\cdot g)(3)\)
\((f\cdot g)(3)\) is the product of \(f(3)\) and \(g(3)\). \[(f\cdot g)(3) = f(3) \cdot g(3) = -9 \cdot 5 = -45\]
06
Calculate \((f/g)(3)\)
\((f/g)(3)\) is the quotient of \(f(3)\) by \(g(3)\). Ensure that \(g(3)\) is not zero to avoid division by zero.\[(f/g)(3) = \frac{f(3)}{g(3)} = \frac{-9}{5} = -1.8\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
evaluating functions
Evaluating functions is a fundamental step in understanding how functions work. In our exercise, we were given two functions:
In the exercise, we substituted \( x = 3 \) into each function:
- \( f(x) = -x^2 \)
- \( g(x) = 2x - 1 \)
In the exercise, we substituted \( x = 3 \) into each function:
- \( f(3) = -(3)^2 = -9 \)
- \( g(3) = 2(3) - 1 = 6 - 1 = 5 \)
function addition
Function addition involves summing the values of two functions for a given input. Once we've evaluated the functions individually, we can add them together.
To calculate the result for \((f+g)(3)\), we used:
To calculate the result for \((f+g)(3)\), we used:
- \( (f+g)(3) = f(3) + g(3) \)
- \( = -9 + 5 \)
- \( = -4 \)
function subtraction
Function subtraction is similar to function addition, but instead we find the difference between two functions. Using the same evaluated values as in the addition step, we performed subtraction:
- \( (f-g)(3) = f(3) - g(3) \)
- \( = -9 - 5 \)
- \( = -14 \)
function multiplication
Function multiplication involves multiplying the values of two functions for a given input. Once you've evaluated the functions, multiplying their results is straightforward. In our exercise, the multiplication was done as follows:
- \( (f \cdot g)(3) = f(3) \cdot g(3) \)
- \( = -9 \cdot 5 \)
- \( = -45 \)
function division
Function division involves dividing the value of one function by the other for the same input. A crucial point in function division is ensuring that the divisor (in this case, the function \(g(x)\)) does not equal zero to avoid undefined operations.
For \((f/g)(3)\), the process was:
For \((f/g)(3)\), the process was:
- \( (f/g)(3) = \frac{f(3)}{g(3)} \)
- \( = \frac{-9}{5} \)
- \( = -1.8 \)
