Problem 30 - Precalculus

Question

Use \(f(x)=-2 x, g(x)=\sqrt{x}\) and \(h(x)=|x|\) to find and simplify expressions for the following functions and state the domain of each using interval notation. $$(f \circ g \circ h)(x)$$

Step-by-Step Solution

Answer

\((f \circ g \circ h)(x) = -2\sqrt{|x|}\) with domain \((-infty, +infty)\).

Step 1: Understand the Composition

The composition \((f \circ g \circ h)(x)\) means we need to apply \(h(x)\) first, then \(g(x)\), and finally \(f(x)\). This can be expressed as \(f(g(h(x)))\).

Step 2: Apply \(h(x)\)

The function \(h(x) = |x|\) takes a number \(x\) and outputs its absolute value. Therefore, \(h(x) = |x|\).

Step 3: Apply \(g(x)\)

The function \(g(x) = \sqrt{x}\) is applied to the output of \(h(x)\). So we substitute \(h(x)\) into \(g(x)\), which gives \(g(h(x)) = g(|x|) = \sqrt{|x|}\). This means we take the square root of the absolute value of \(x\).

Step 4: Apply \(f(x)\)

The function \(f(x) = -2x\) is applied to the output of \(g(h(x))\). So, we substitute \(g(h(x))\) into \(f(x)\), resulting in \(f(g(h(x))) = f(\sqrt{|x|}) = -2 \times \sqrt{|x|}\).

Step 5: State the Domain

The domain is determined by the inner-most function with the most restrictions, in this case, \(g(x) = \sqrt{x}\). Since square roots are only defined for non-negative numbers, and since we are taking the square root of \(|x|\), the domain includes all real numbers because \(|x|\) is always non-negative. Therefore, the domain is \((-infty, +infty)\).

Key Concepts

Domain of Functions

When discussing functions, the 'domain' is just a fancy term for all the possible input values (usually called \(x\)) that you can plug into a function without breaking any math rules.
For the function \(g(x) = \sqrt{x}\), you can't take the square root of a negative number (in the realm of real numbers), so its domain is all non-negative numbers.
For absolute value, \(h(x) = |x|\), the domain is all real numbers because you can take the absolute value of any number.

• For \(f(x) = -2x\), the domain is also all real numbers.

When combining functions, as in \((f \circ g \circ h)(x)\), the domain is the most restricted of all the functions involved. Here, the operations result in \(\sqrt{|x|}\), making \(|x|\) always non-negative, thus allowing \(g(x)\) to be applied to any real number.
So, the domain of \((f \circ g \circ h)(x)\) is written in interval notation as \((-\infty, +\infty)\).

Absolute Value

Absolute value is like a magic function that turns every number into its positive version, or keeps it the same if it's already positive!
Represented by the bars \(|x|\), it literally measures the 'distance' of a number \(x\) from zero on a number line.
So, \(|-3| = 3\) because we don't worry about the minus; it's just 3 steps away from zero.

• If \(x\) is positive, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\) ( which flips it positive).

In function compositions, using \(|x|\) ensures that whatever your input is—from negative to positive—the operations that follow deal with a non-negative value.
This becomes crucial for functions like square roots, as only non-negative numbers can be used for the square root operation in real math.

Square Root Functions

The square root is like the inverse of squaring a number. It's a bit like asking, “What number squared will give me this original number?” The classic notation for a square root is \(\sqrt{\cdot}\).
Now, \(g(x) = \sqrt{x}\) means for every \(x\), return the value that squared would produce \(x\).

• For example, \(\sqrt{16} = 4\) because \(4^2 = 16\).
• Importantly, \(\sqrt{x}\) is only defined for \(x \geq 0\); you can't find the square root of a negative number in the set of real numbers.

When you wrap \(\sqrt{\cdot}\) around an absolute value function, like \(\sqrt{|x|}\), the outcome is always defined and non-negative since \(|x|\) is non-negative. This makes the square root safe to calculate!

Interval Notation

Interval notation is a simple shorthand used to describe a set of numbers between two endpoints. It's a way of showing which numbers are included (or excluded) in a range.
Instead of writing out "\(-1 < x < 5\)", use \((-1, 5)\). The parentheses mean the endpoints themselves aren't included.

• If an endpoint is included, we use square brackets like so: \([-1, 5)\).
• If it goes on forever negatively or positively, use \(-\infty\) and \(+\infty\).

For example, the domain \((-\infty, +\infty)\) for our function composition \( (f \circ g \circ h)(x) \) tells us that all real numbers are good inputs.
This notation is concise and efficient, making it easier to visually grasp and communicate ranges through math problems.