Problem 22 - Contemporary Precalculus

Question

Find the rule of the function \(f \circ g,\) the domain of \(f \circ g,\) the rule of \(g \cdot f,\) and the domain of \(g \circ f\) $$f(x)=1 / x, \quad g(x)=\sqrt{x}$$

Step-by-Step Solution

Answer

Answer: The rule for the composite function \(f \circ g\) is \((f \circ g)(x)=\frac{1}{\sqrt{x}}\) with domain \(x > 0\). The rule for the composite function \(g \circ f\) is \((g \circ f)(x)=\sqrt{\frac{1}{x}}\) with domain \(x > 0\).

Step 1: Find the rule of \(f \circ g\)

To find the rule of the composite function \(f \circ g\), evaluate \(f(g(x))\): $$ (f \circ g)(x)=f(g(x))=f(\sqrt{x}) $$ Since \(f(x)=1/x\), replace the input in the function \(f\) with \(\sqrt{x}\): $$ f(\sqrt{x})=\frac{1}{\sqrt{x}} $$ Therefore, the rule for \(f \circ g\) is \((f \circ g)(x)=\frac{1}{\sqrt{x}}\).

Step 2: Find the domain of \(f \circ g\)

To find the domain of the function \(f \circ g\), consider where the function \(g\) is defined and where the composite function \(f(g(x))\) is defined. Since \(g(x)=\sqrt{x}\), \(g\) is defined for all \(x \geq 0\). Moreover, the function \(f(x)=1/x\) is defined for all \(x \neq 0\). Therefore, the domain of \(f \circ g\) is all values of \(x\) such that \(x > 0\). The domain of \(f \circ g\) is: \(x > 0\).

Step 3: Find the rule of \(g \circ f\)

To find the rule of the composite function \(g \circ f\), evaluate \(g(f(x))\): $$ (g \circ f)(x)=g(f(x))=g\left(\frac{1}{x}\right) $$ Since \(g(x)=\sqrt{x}\), replace the input in the function \(g\) with \(\frac{1}{x}\): $$ g\left(\frac{1}{x}\right)=\sqrt{\frac{1}{x}} $$ Therefore, the rule for \(g \circ f\) is \((g \circ f)(x)=\sqrt{\frac{1}{x}}\).

Step 4: Find the domain of \(g \circ f\)

To find the domain of the function \(g \circ f\), consider where the function \(f\) is defined and where the composite function \(g(f(x))\) is defined. Since \(f(x)=1/x\), \(f\) is defined for all \(x \neq 0\). Moreover, the function \(g(x)=\sqrt{x}\) is defined for all \(x \geq 0\). Therefore, we must consider the values of \(x\) for which \(f(x)\) is non-negative. Since \(f(x)\) is a reciprocal function, it will be non-negative for all \(x > 0\). The domain of \(g \circ f\) is: \(x > 0\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (usually represented by \(x\)) that the function can accept without any mathematical errors.
Understanding the domain requires you to ensure that the operations inside the function are defined and do not involve division by zero or taking the square root of a negative number, for instance.

• For \(f(x)=\frac{1}{x}\): The function \(f\) is defined for all \(xeq0\), because dividing by zero is undefined in mathematics.
• For \(g(x)=\sqrt{x}\): The function \(g\) is defined for all \(x\geq0\), since we cannot take the square root of a negative number in the realm of real numbers.

To find the domain of a composite function like \(f \circ g\), you need both sets of domain restrictions to hold. For example, \(f \circ g\) is defined only when \(g(x)\) outputs values that are valid inputs for \(f\). Hence, the domain of \(f \circ g(x)=\frac{1}{\sqrt{x}}\) is all \(x>0\), since \(x=0\) would make \(\frac{1}{\sqrt{x}}\) undefined.

Composite Functions

Composite functions are created when the output of one function becomes the input to another. We denote this with \((f \circ g)(x) = f(g(x))\). This situation is like processing data through two machines where each machine applies its own operation on the data.
To find a composite function

• First, compute the inner function, \(g(x)\).
• Then, take this result and substitute it into the outer function, \(f(x)\).

Knowing the composite function rule helps describe how changes in \(x\) are transformed through both functions. For instance, in \(f \circ g\), first, replace \(x\) with \(\sqrt{x}\) to get \(g(x)=\sqrt{x}\), and then apply \(f\) on this result: \(f(\sqrt{x})=\frac{1}{\sqrt{x}}\). Each step transforms \(x\) in a planned sequence.

Square Root Function

The square root function, denoted by \(\sqrt{x}\), is a specific type of function that returns non-negative values.
This means if you input \(x\) into the square root function, the result is \(y\) such that \(y^2 = x\). This operation only makes sense within the set of non-negative real numbers.
Here are some key points about the square root function:

• Defined only for \(x\geq0\).
• The output, \(\sqrt{x}\), is always non-negative even for positive \(x\).
• The growth rate of \(\sqrt{x}\) decreases as \(x\) increases, meaning it takes bigger inputs to make significant increases in the output.

When used inside composite functions, like \(f \circ g\) or \(g \circ f\), it's crucial to keep in mind its domain restrictions. Such limitations impact the overall domains when composed with other functions, ensuring no undefined operations occur.