Problem 4 - College Algebra

Question

Let \(f(x)=\sqrt{x-8}\) and \(g(x)=\sqrt{x+5},\) find: a. \((f+g)(x)\) b. the domain of \(f+g .\)

Step-by-Step Solution

Answer

The function \(f+g(x)\) = \( \sqrt{x-8} + \sqrt{x+5}\) and the domain of \(f+g\) is \(x \geq 8\).

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

This step involves adding the two functions together. \(f(x) + g(x) = \sqrt{x-8} + \sqrt{x+5}\). This is the function \(f+g(x)\).

Step 2: Find the Domain of (f+g)(x)

The domains of the functions \(f(x)\) and \(g(x)\) must both be valid, since the function \(f+g(x)\) combines the two. Therefore, it is important to isolate each function and find the values of \(x\) that make the values under the square roots non-negative. So, \(x-8 \geq 0\) and \(x+5 \geq 0\) are the constraints for the domains for \(f(x)\) and \(g(x)\), respectively. Solving these inequalities for \(x\) gives \(x \geq 8\) and \(x \geq -5\), respectively. When combined, the shared domain should only consider values from the highest lower bound, which in this case is 8. Therefore, the domain of \(f+g\) is \(x \geq 8.\)

Key Concepts

Domain of a Function

In mathematics, the domain of a function is crucial for understanding where the function works. It's like a set of all possible starting points. When you look at a function, you're often trying to understand all the values of \(x\) that you can plug in without breaking the function. Each function sometimes has limits on what \(x\) can be to avoid issues like division by zero or negative roots for square functions.

For example, when we say "find the domain of \(f(x) = \sqrt{x-8}\)", we're searching for all the values of \(x\) that make the expression inside the square root valid, or in this case, non-negative. This leads us to solving inequalities that inform us about the permissible values for \(x\). Always remember, the number under a square root (radicand) has to be \(0\) or positive for real numbers.

• Identify any restrictions (such as forbidden values or negative numbers under a square root)
• Solve for \(x\) to pinpoint all permissible values
• The result gives the domain of the function, often outlined in interval notation itself or as an inequality

Square Root Function

The square root function, denoted as \(f(x) = \sqrt{x}\), plays a special role in algebra. It involves finding the original number that was squared to arrive at \(x\), effectively undoing a squaring operation. It's important to note that you can't take the square root of negative numbers in the set of real numbers. That's why the radicand (the number inside the square root) must always be non-negative.

In practical terms:

• If \(f(x) = \sqrt{x-8}\), \(x-8\) must be \(\geq 0\)
• This tells us \(x\) must be \(\geq 8\)
• For each function you consider, set its radicand \(\geq 0\) to find valid \(x\)

Remember, square root functions create smooth curves, starting from the point where the radicand is zero and moving onwards. In graphs, these functions operate only in the first quadrant because they do not go below the \(x\)-axis in real number calculations.

Inequalities

Inequalities are a mathematical expression used to compare two values or expressions, stating that one is less than or greater than the other. In our context, inequalities help determine the domain of functions, particularly when dealing with square roots. They form a condition that needs to be satisfied by \(x\) so that no illegal operations occur in the function.

For example, in solving \(x-8 \geq 0\) and \(x+5 \geq 0\), you isolate \(x\) to find the sets of viable values:

• The solution to \(x-8 \geq 0\) is \(x \geq 8\)
• The solution to \(x+5 \geq 0\) is \(x \geq -5\)

When combining these solutions, you look for the intersection of \(x\) values, meaning the overlap that satisfies both inequalities. In this case, it's the larger of the two lower bounds: \(x \geq 8\). By solving inequalities, we effectively decide the region or set where a function is defined and operates validly.