Problem 29 - A Graphical Approach to Precalculus with Limits

Question

For each pair of fimetions, (a) find ( \(f+g)(x),(f-g)(x),\) and \((f g)(x) ;\) (b) give the domains of the functions in part (a); (c) find \(\frac{t}{x}\) and give its domain; (d) find \(f \circ g\) and give is domain: and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=\sqrt{x^{2}+3}, g(x)=x+1$$

Step-by-Step Solution

Answer

(f+g)(x) = \sqrt{x^2+3} + x +1; (f-g)(x) = \sqrt{x^2+3} - x -1; (f*g)(x) = \sqrt{x^2+3}(x+1); Dom: all x; (f/g)(x) und. at x = -1; (f∘g)(x)=\sqrt{x^2+2x+4}, all x; (g∘f)(x) = \sqrt{x^2+3}+1, all x.

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

To find \((f+g)(x)\), add the functions \(f(x)\) and \(g(x)\):\[f(x) + g(x) = \sqrt{x^2 + 3} + (x + 1) = \sqrt{x^2 + 3} + x + 1.\]

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

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\):\[f(x) - g(x) = \sqrt{x^2 + 3} - (x + 1) = \sqrt{x^2 + 3} - x - 1.\]

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

To find \((f\cdot g)(x)\), multiply \(f(x)\) by \(g(x)\):\[ (f \cdot g)(x) = \sqrt{x^2 + 3} \cdot (x + 1). \]

Step 4: Find Domains of (f+g)(x), (f-g)(x), (f*g)(x)

Function \(f(x) = \sqrt{x^2 + 3}\) is defined for all real numbers since the expression under the square root, \(x^2 + 3\), is always positive. Therefore, the domain of \(f(x)\) and \(g(x) = x + 1\) (which is also all real numbers) yields a combined domain of all real numbers for \((f+g)(x), (f-g)(x),\) and \((f\cdot g)(x)\).

Step 5: Compute and Find Domain of (f/g)(x)

To find \(\frac{f}{g}(x)\), divide \(f(x)\) by \(g(x)\):\[\frac{f}{g}(x) = \frac{\sqrt{x^2 + 3}}{x + 1}.\]The domain includes all real numbers except where \(g(x) = 0\), which occurs when \(x = -1\). Therefore, the domain is all real numbers except \(x = -1\).

Step 6: Compute f ◦ g and Find Its Domain

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\): \[ f(g(x)) = \sqrt{(x + 1)^2 + 3} = \sqrt{x^2 + 2x + 4}. \]The domain is all real numbers because the expression inside the square root is non-negative for all real \(x\).

Step 7: Compute g ◦ f and Find Its Domain

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\): \[ g(f(x)) = \sqrt{x^2 + 3} + 1. \]The domain is all real numbers, as \(f(x)\) is defined for all \(x\).

Key Concepts

Domain of Functions

The domain of a function refers to the complete set of possible input values (usually represented as "x") for which the function is defined. In simpler terms, the domain tells us what values we can plug into the function without breaking any mathematical rules.
Let's consider some important points about domains:

• For functions involving square roots, the expression inside the root must be non-negative. For example, in the function \(f(x) = \sqrt{x^2 + 3}\), the expression \(x^2 + 3\) is always positive (or zero), making the domain of \(f(x)\) all real numbers.


• For rational functions (where you divide one function by another), such as \(f(x) / g(x)\), the denominator must not be zero. So, we exclude any values of \(x\) that make the denominator zero from the domain.


• Polynomial functions like \(g(x) = x + 1\) usually have domains of all real numbers since there are no restrictions like division by zero or negative square roots.

Function Composition

Function composition involves applying one function to the results of another. The notation \((f \circ g)(x)\) means you first apply \(g(x)\), then apply \(f(x)\) to the result of \(g(x)\). This allows for chaining functions together to create new, more complex functions.
To understand function composition fully, note:

• The composite function \((f \circ g)(x)\) is read as "\(f\) of \(g\) of \(x\)".


• In the problem example, \((f \circ g)(x) = \sqrt{(x + 1)^2 + 3}\). Here, after \(g(x) = x + 1\), you apply \(f(x) = \sqrt{x^2 + 3}\), making sure the domain holds true throughout the composition.


• Conversely, \((g \circ f)(x) = \sqrt{x^2 + 3} + 1\) means you first apply \(f(x)\), then \(g(x)\).

Knowing the order is crucial since \(f \circ g\) and \(g \circ f\) may yield different results.

Square Roots

Square roots are an important concept in many mathematical functions. The square root of a number \(a\) is a value that, when multiplied by itself, gives \(a\). It's usually denoted as \(\sqrt{a}\).
Key points about square roots:

• The expression inside a square root (called the radicand) must be non-negative for the result to be a real number. For \(f(x) = \sqrt{x^2 + 3}\), \(x^2 + 3\) is always non-negative, allowing all real numbers as inputs.


• When dealing with square roots in a function's definition, always check if there are any restrictions on the values that \(x\) can take based on the radicand.

The concept extends to the operations between functions, ensuring that computations always involve real numbers.

Real Numbers

Real numbers include all rational and irrational numbers. Essentially, they are the numbers we are most familiar with in everyday number tasks, like counting and measuring.
In the context of functions, real numbers play a vital role:

• When determining a function's domain, we often say it's all real numbers, meaning any real number can be used as an input without causing undefined behavior in the function.


• In our problem, both \(f(x) = \sqrt{x^2 + 3}\) and \(g(x) = x + 1\) have a domain of all real numbers.


• Being comfortable with real numbers is key to understanding function operations and their possible domains, ensuring that all calculations yield valid results.

Real numbers form the foundation upon which we explore more complex mathematical concepts.