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Function addition is a basic operation that involves combining two functions into a single function by adding their expressions. In this case, for the functions given as \(f(x) = x + 4\) and \(g(x) = 5x - 2\), the sum of these functions is denoted as \((f+g)(x)\). To perform this addition, you simply add the expressions of \(f(x)\) and \(g(x)\):

• \((f+g)(x) = (x + 4) + (5x - 2)\)

Next, combine the like terms to simplify the expression. Like terms are those that have the same variable raised to the same power. Here, the terms with \(x\) are \(x\) and \(5x\), and the constant terms are \(4\) and \(-2\).

• Combine \(x\) and \(5x\) to get \(6x\).
• Combine \(4\) and \(-2\) to get \(2\).

So, the function resulting from addition is \((f+g)(x) = 6x + 2\). This new function now represents all the outputs of the combined original functions for the same input \(x\).

Function subtraction involves taking one function away from another, resulting in a new function. Given the functions \(f(x) = x + 4\) and \(g(x) = 5x - 2\), the difference is expressed as \((f-g)(x)\). To subtract, you take the expression for \(g(x)\) and subtract it from \(f(x)\):

• \((f-g)(x) = (x + 4) - (5x - 2)\)

Now, apply the subtraction across each term:

• The \(x\) term becomes \(x - 5x\), which simplifies to \(-4x\).
• The constant \(4\) minus \(-2\) becomes \(4 + 2 = 6\).

Thus, the function after subtraction is \((f-g)(x) = -4x + 6\). This operation effectively shows how the outputs of \(g(x)\) are subtracted from those of \(f(x)\) for each input \(x\).

Function multiplication involves creating a new function by multiplying two existing functions together. For \(f(x) = x + 4\) and \(g(x) = 5x - 2\), their product is \((f \cdot g)(x)\). To find this, multiply the two function expressions:

• \((f \cdot g)(x) = (x + 4)(5x - 2)\)

To simplify, use the distributive property, which involves multiplying each term in the first function by each term in the second function. Follow these steps:

• Multiply \(x\) by \(5x\) to get \(5x^2\).
• Multiply \(x\) by \(-2\) to get \(-2x\).
• Multiply \(4\) by \(5x\) to get \(20x\).
• Multiply \(4\) by \(-2\) to get \(-8\).

Combine the like terms, \(-2x\) and \(20x\):

• \(5x^2 + 18x - 8\)

This expression, \(5x^2 + 18x - 8\), represents the function which is the product of \(f\) and \(g\).

Function division creates a new function by dividing one function by another. For the provided functions, \(f(x) = x + 4\) and \(g(x) = 5x - 2\), you get the quotient function \(\left(\frac{f}{g}\right)(x)\) by dividing \(f(x)\) by \(g(x)\):

• \(\left(\frac{f}{g}\right)(x) = \frac{x + 4}{5x - 2}\)

In this division, \(f(x)\) becomes the numerator and \(g(x)\) the denominator. This expression is a rational function because it indicates how one function's outputs scale compared to another's for the same input \(x\). Important to note is that this rational expression is in its simplest form, meaning no further simplification is possible without specific values for \(x\). Each value of \(x\) should, however, consider that \(g(x)\) must not be zero, as division by zero is undefined.