Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find \(f \circ g, g \circ f,\) and the domain of \(f \circ g .\) (b) Use a graphing utility to graph \(f \circ g\) and \(g \circ f .\) Determine whether \(f \circ g=g \circ f.\) $$f(x)=\sqrt{x+4}, \quad g(x)=x^{2}$$

Step-by-Step Solution

Answer

f(g(x)) = \(\sqrt{x²+4}\) with domain \(-2 \leq x \leq 2\), g(f(x)) = \(x + 4\) with domain \(x \geq -4\). f(g(x)) does not equal g(f(x)).

Step 1: Finding f(g(x))

To find f(g(x)) substitute x² from g(x) into f(x): \(f(g(x)) = f(x²) = \sqrt{x²+4}\). The domain of f(g(x)) is all x such that \(-2 \leq x \leq 2\) since square root function requires its argument to be nonnegative.

Step 2: Finding g(f(x))

To find g(f(x)) substitute \sqrt{x+4} from f(x) into g(x): \(g(f(x)) = g(\sqrt{x+4}) = (\sqrt{x+4})² = x + 4\). The domain of g(f(x)) is \(x \geq -4\) since square root function requires its argument to be nonnegative.

Step 3: Compare f(g(x)) and g(f(x))

Observe that f(g(x)) = \(\sqrt{x²+4}\) and g(f(x)) = \(x + 4\) are not the same. Therefore, f(g(x)) does not equal g(f(x)).

Step 4: Graphing and Comparing f(g(x)) and g(f(x))

To graph the functions, insert the functions into the graphing utility and graph. By observation of the graphs you will also see that these two functions are not the same. Notice for graphing you just need to input x values from their domains.

Key Concepts

Domain of a function

Understanding the domain of a function is crucial in mathematics as it defines the set of input values (usually "x" values) for which the function is defined. For any given function, the domain can be limited by factors such as division by zero or the square root of a negative number. In this context, let's consider composite functions like \(f \circ g\) and \(g \circ f\).

• For \(f(g(x)) = \sqrt{x^2 + 4}\), the expression under the square root, \(x^2 + 4\), must be non-negative. Since \(x^2\) is always non-negative and \(4\) is a positive number, this function is valid for all real numbers. However, here we've specified that due to the nature of function composition, the domain is actually limited to \(-2 \leq x \leq 2\). This restriction comes from ensuring all parts of the composite function are valid.
• For \(g(f(x)) = x + 4\), what we initially need is that the square root function \(\sqrt{x + 4}\) is defined. Since the expression under the square root must be non-negative, \(x + 4 \geq 0\) leads to \(x \geq -4\).

Graphing functions

Graphing functions helps us visualize the relationship they describe. Graphs display how outputs (often "y") change with different inputs ("x" values). When graphing composite functions like \(f \circ g\) and \(g \circ f\), pay attention to their domains and codomains.

• \(f(g(x))\) or \(\sqrt{x^2 + 4}\) results in a graph that is only defined between \(-2\) and \(2\). This represents a range of real numbers generated by substituting "x" in \(x^2 + 4\).
• Conversely, graphing \(g(f(x)) = x + 4\) results in a linear graph \(y = x + 4\). It's a line that continues on forever from \(x \geq -4\).

By using graphing utilities, you can enter these functions and observe their behaviors visually, confirming that these two are not equivalent graphs as they don't overlap or mirror each other.

Square root function

The square root function is a pivotal concept in algebra, symbolized by \(\sqrt{\phantom{x}}\). A function of the form \(f(x) = \sqrt{x}\) assumes only non-negative arguments.

• The domain of a square root function is limited to values of "x" that make the expression inside the square root non-negative, i.e., \(x \geq 0\). For example, the square root function \(\sqrt{x + 4}\) has a domain of \(x \geq -4\) as it requires the inside to remain \(\geq 0\).
• When analyzing functions like \(f(x) = \sqrt{x + 4}\), remember that this shifts the graph of the basic square root function \(f(x) = \sqrt{x}\) horizontally to the left by 4 units.

The square root function produces only non-negative results, meaning it only appears on and above the x-axis in a graph. Its simple structure helps in understanding complex functions when they include square roots.

Quadratic function

Quadratic functions are polynomials of the form \(f(x) = ax^2 + bx + c\), where "a", "b", and "c" are constants. They produce a characteristic "U" shaped curve known as a parabola.

• In our example, \(g(x) = x^2\) is a basic quadratic function with "a" = 1, "b" = 0, and "c" = 0. The vertex of this parabola is at the origin, \((0,0)\), and it opens upwards.
• Quadratics are defined for all real numbers, meaning their domain is \((-\infty, \infty)\).

Quadratic functions are symmetrical about their vertex, providing numerous analytical properties. Their graph informs us about the function's roots, and it's visually pleasing for identifying intersections when examining composite functions such as \(f \circ g\) and \(g \circ f\). Understanding these properties helps simplify the analysis of more complex expressions.