Problem 6
Question
Find \(f+g, f-g, f g,\) and \(\frac{f}{g}\). Determine the domain for each function. $$f(x)=3 x-4, g(x)=x+2$$
Step-by-Step Solution
Verified Answer
The results of the operations are: \(f(x) + g(x) = 4x - 2\), \(f(x) - g(x) = 2x - 6\), \(f(x) \cdot g(x) = 3x^2 + 2x - 8\), and \(\frac{f(x)}{g(x)} = \frac{(3x - 4)}{(x+2)}\). The domain of the first three results is all real numbers. For the last operation, the domain is all real numbers except -2.
1Step 1: Addition of Functions
The addition of the two functions \(f(x)\) and \(g(x)\) is obtained by adding the corresponding expressions for each x. \(f(x) + g(x) = (3x - 4) + (x+2) = 4x - 2\). The domain of this function is all real numbers since there are no restrictions on x.
2Step 2: Subtraction of Functions
The subtraction of the two functions \(f(x)\) and \(g(x)\) is obtained by subtracting the corresponding expressions for each x. \(f(x) - g(x) = (3x - 4) - (x+2) = 2x - 6\). The domain of this function is all real numbers as well.
3Step 3: Multiplication of Functions
The multiplication of the two functions \(f(x)\) and \(g(x)\) is obtained by multiplying the corresponding expressions for each x. \(f(x) \cdot g(x) = (3x - 4) \cdot (x+2) = 3x^2 + 2x - 8\). The domain of this function is all real numbers as well.
4Step 4: Division of Functions
The division of the two functions \(f(x)\) and \(g(x)\) is obtained by dividing the corresponding expressions for each x. It's necessary to make sure that the denominator \(g(x) = x + 2\) is not equal to 0, because it's undefined there. So, for \(\frac{f(x)}{g(x)}\), we need to solve the equation \(x + 2 = 0\). The solution is \(x = -2\). Therefore, \(x\) can be any real number except for -2. So, the domain of \(\frac{f(x)}{g(x)} = \frac{(3x - 4)}{(x+2)}\) is all real numbers except -2.
Key Concepts
Addition of FunctionsSubtraction of FunctionsMultiplication of FunctionsDivision of FunctionsDomain of Functions
Addition of Functions
Understanding the addition of functions is like combining two recipes to create a new dish. In algebra, when we have two functions, say f(x) and g(x), their sum f(x) + g(x) is simply a matter of adding their respective formulas together. For the functions f(x) = 3x - 4 and g(x) = x + 2, adding these gives us a new function h(x) = (3x - 4) + (x + 2) = 4x - 2.
This new function, h(x), can take any real number as input, meaning there's no x-value at which the function doesn't work. We say that its domain is all real numbers, represented as \( -\infty, \infty \).
This new function, h(x), can take any real number as input, meaning there's no x-value at which the function doesn't work. We say that its domain is all real numbers, represented as \( -\infty, \infty \).
- Why is the domain all real numbers? Because there are no divisions by zero or square roots of negative numbers, which are common deal breakers in defining domains.
Subtraction of Functions
When we subtract one function from another, it's like taking ingredients out of a recipe to see how it changes. With our example functions f(x) and g(x), subtraction gives us another new function f(x) - g(x) = (3x - 4) - (x + 2) = 2x - 6.
Just like with addition, the domain of the resulting function from subtraction is all real numbers. That's because there's nothing in the formula 2x - 6 to prevent any real number from being plugged into x. So, no matter what value of x you choose, you will always get a real number out, making the domain \( -\infty, \infty \).
Just like with addition, the domain of the resulting function from subtraction is all real numbers. That's because there's nothing in the formula 2x - 6 to prevent any real number from being plugged into x. So, no matter what value of x you choose, you will always get a real number out, making the domain \( -\infty, \infty \).
Multiplication of Functions
Multiplying functions works like layering flavors in a complex dish. For our given functions, we multiply the algebraic expressions: f(x) \times g(x) = (3x - 4) \times (x + 2).
To do this, we distribute every term in the first function by every term in the second, resulting in f(x) \times g(x) = 3x^2 + 6x - 4x - 8 = 3x^2 + 2x - 8.
Just like addition and subtraction, the domain remains all real numbers here because our result doesn't introduce any mathematical restrictions such as division by zero. Thus, the domain of this new multiplied function is also \( -\infty, \infty \).
To do this, we distribute every term in the first function by every term in the second, resulting in f(x) \times g(x) = 3x^2 + 6x - 4x - 8 = 3x^2 + 2x - 8.
Just like addition and subtraction, the domain remains all real numbers here because our result doesn't introduce any mathematical restrictions such as division by zero. Thus, the domain of this new multiplied function is also \( -\infty, \infty \).
Division of Functions
Dividing one function by another can sometimes be complex, similar to tweaking the ratio of ingredients in a recipe for the perfect flavor balance. Here, when we divide f(x) by g(x), we get: \( \dfrac{f(x)}{g(x)} = \dfrac{3x - 4}{x + 2} \).
However, we must be careful. Division has a strict rule: you cannot divide by zero. Therefore, we must ensure the denominator isn't zero at any point. In our case, x + 2 equals zero when x is -2, which we exclude from the domain. As a result, the domain for this division function is all real numbers except for -2, which we notate as: \(( -\infty, -2) \cup (-2, \infty) \).
However, we must be careful. Division has a strict rule: you cannot divide by zero. Therefore, we must ensure the denominator isn't zero at any point. In our case, x + 2 equals zero when x is -2, which we exclude from the domain. As a result, the domain for this division function is all real numbers except for -2, which we notate as: \(( -\infty, -2) \cup (-2, \infty) \).
- This is crucial because if we don't identify this exclusion, we may encounter undefined expressions, which do not make sense in the quantifiable world of mathematics.
Domain of Functions
Imagine having a set of tools, but you need to know which ones are usable for a particular job. The domain of a function tells us exactly which 'tools'—or values of x—we can use.
To find the domain, we look for values of x that don't work, usually due to division by zero or taking the root of a negative in the context of real numbers.
For our functions, both f(x) and g(x) have straightforward expressions without these issues, implying their individual domains are all real numbers. But as we've seen with division, combining functions can introduce restrictions like excluding x = -2 to avoid division by zero.
Understanding domain is essential in ensuring all operations are valid within the constraints of the functions and real numbers. Remember that it is the set of possible inputs that give valid outputs, and mathematicians really don't like 'undefined' on their plate.
To find the domain, we look for values of x that don't work, usually due to division by zero or taking the root of a negative in the context of real numbers.
For our functions, both f(x) and g(x) have straightforward expressions without these issues, implying their individual domains are all real numbers. But as we've seen with division, combining functions can introduce restrictions like excluding x = -2 to avoid division by zero.
Understanding domain is essential in ensuring all operations are valid within the constraints of the functions and real numbers. Remember that it is the set of possible inputs that give valid outputs, and mathematicians really don't like 'undefined' on their plate.
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