Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=2 x+5, \quad g(x)=x^{2}-9$$

Step-by-Step Solution

Answer

\((f+g)(x) = x^2 + 2x - 4\), \((f-g)(x) = -x^2 + 2x + 14\), \((f*g)(x) = 2x^3 -18x + 5x^2 - 45\), \((f/g)(x) = (2x + 5) / (x^2 - 9)\), the domain of \(f/g\) is all real numbers except \(x = 3\) and \(x = -3\)

Step 1: Finding (f+g)(x)

The sum of two functions \(f\) and \(g\), denoted as \((f+g)(x)\), is obtained by adding the mathematical expressions of the two functions. Hence, \((f+g)(x) = f(x) + g(x)\) So, \((f+g)(x) = 2x + 5 + x^2 - 9 = x^2 + 2x - 4\)

Step 2: Finding (f-g)(x)

The difference between two functions \(f\) and \(g\), denoted as \((f-g)(x)\), is obtained by subtracting the mathematical expression of \(g(x)\) from \(f(x)\). So, \((f-g)(x) = f(x) - g(x)\) Therefore, \((f-g)(x) = 2x + 5 - (x^2 - 9) = -x^2 + 2x + 14\)

Step 3: Finding (f*g)(x)

The product of two functions \(f\) and \(g\), denoted as \((f*g)(x)\), is obtained by multiplying the mathematical expressions of the two functions. So, \((f*g)(x) = f(x) * g(x)\) Hence, \((f*g)(x) = (2x + 5)*(x^2 - 9) = 2x^3 -18x + 5x^2 - 45\)

Step 4: Finding (f/g)(x)

The division of two functions \(f\) and \(g\), denoted as \((f/g)(x)\), is obtained by dividing the mathematical expression of \(f(x)\) by \(g(x)\). So, \((f/g)(x) = f(x) / g(x)\) Hence, \((f/g)(x) = (2x + 5) / (x^2 - 9)\)

Step 5: Finding the domain of (f/g)(x)

The domain of a function includes all real numbers except those for which the denominator is zero. Here, the denominator of \((f/g)(x)\) is \(x^2 - 9\), so it is zero when \(x = 3\) and \(x = -3\). Hence, the domain of \(f/g\) is all real numbers except \(x = 3\) and \(x = -3\)

Key Concepts

Function Addition

Function addition, expressed as \((f+g)(x)\), involves adding two functions together. You simply add each corresponding term from the two formulas. For instance, given \(f(x) = 2x + 5\) and \(g(x) = x^2 - 9\), the sum \((f+g)(x)\) would be \(f(x) + g(x) = 2x + 5 + x^2 - 9\). Simplifying this, you combine like terms to get \((f+g)(x) = x^2 + 2x - 4\). This process is straightforward, as it revolves around simple addition. Just remember to organize the terms by degree for a clear and neat expression.

• Add corresponding terms from each function.
• Simplify by combining like terms.

Function Subtraction

Function subtraction is similar to addition but involves subtracting the expressions instead. The expression for subtraction is given as \((f-g)(x)\). To perform this, you subtract the function \(g(x)\) from \(f(x)\). Using our functions \(f(x) = 2x + 5\) and \(g(x) = x^2 - 9\), the subtraction would look like this: \((f-g)(x) = f(x) - g(x) = 2x + 5 - (x^2 - 9)\). It is crucial to distribute the negative sign across all terms in \(g(x)\) which gives \(-x^2 + 2x + 14\) after simplifying.

• Subtract \(g(x)\)'s terms from \(f(x)\)'s terms.
• Distribute the negative sign correctly.

Function Multiplication

Multiplying functions involves taking the expression of one function and multiplying it by the other. This is noted as \((f*g)(x)\). When multiplying \(f(x) = 2x + 5\) and \(g(x) = x^2 - 9\), you perform a distributive multiplication between the sets of terms. This means each term in \(f(x)\) is multiplied by each term in \(g(x)\). So, \((f*g)(x) = (2x + 5)(x^2 - 9)\) expands into \((2x)(x^2) + (2x)(-9) + (5)(x^2) + (5)(-9)\). After simplifying, the result is \(2x^3 + 5x^2 - 18x - 45\). Always remember to combine like terms at the end.

• Distribute each term of \(f(x)\) across \(g(x)\).
• Simplify by combining like terms.

Function Division

Dividing functions, represented as \((f/g)(x)\), involves dividing one function's expression by another's. Here, \((f/g)(x)\) is calculated by taking \(f(x)\) and dividing it by \(g(x)\) so \((f/g)(x) = \frac{2x + 5}{x^2 - 9}\). Division is generally straightforward, but do note it imposes restrictions on the domain because the denominator cannot be zero.

• Ensure the denominator does not equal zero.
• Set up the division expression and simplify if possible.

Domain of a Function

The domain of a function is the set of all possible inputs (x-values) that result in real numbers for the function's output. When dealing with function division, special attention is required to ensure that the denominator does not equate to zero, as this renders the function undefined.
Considering \((f/g)(x) = \frac{2x + 5}{x^2 - 9}\), we set the denominator, \(x^2 - 9 = 0\). Solving \(x^2 = 9\), we get \(x = 3\) and \(x = -3\) which are excluded from the domain. Thus, the domain of \(f/g\) is all real numbers except \(+3\) and \(-3\).

• Identify values that make the denominator zero and exclude them.
• State the domain as all real numbers except those excluded values.