Problem 9 - Calculus

Question

For the functions \(f\) and \(g\) find (a) \(f(g(1))\) (b) \(g(f(1))\) (c) \(f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=\sqrt{x+4}, g(x)=x^{2}$$

Step-by-Step Solution

Answer

(a) \(\sqrt{5}\), (b) 5, (c) \(\sqrt{x^2+4}\), (d) \(x+4\), (e) \(t^2\sqrt{t+4}\).

Step 1: Evaluate inner function for (a)

For (a), start by evaluating the inside function first: \(g(1)\).Given \(g(x) = x^2\), plug in \(x = 1\):\[g(1) = 1^2 = 1\]

Step 2: Evaluate outer function for (a)

Now that you have \(g(1)\), substitute into \(f(x)\) to find \(f(g(1))\).Given \(f(x) = \sqrt{x+4}\), plug in \(g(1) = 1\):\[f(g(1)) = f(1) = \sqrt{1+4} = \sqrt{5}\]

Step 3: Evaluate inner function for (b)

For (b), start by evaluating the inside function: \(f(1)\).Given \(f(x) = \sqrt{x+4}\), plug in \(x = 1\):\[f(1) = \sqrt{1+4} = \sqrt{5}\]

Step 4: Evaluate outer function for (b)

Now that you have \(f(1)\), substitute into \(g(x)\) to find \(g(f(1))\).Given \(g(x) = x^2\), plug in \(f(1) = \sqrt{5}\):\[g(f(1)) = (\sqrt{5})^2 = 5\]

Step 5: Compose function expressions for (c)

For (c), substitute \(g(x)\) into \(f(x)\) to find \(f(g(x))\).Given \(g(x) = x^2\) and \(f(x) = \sqrt{x+4}\), substitute:\[f(g(x)) = f(x^2) = \sqrt{x^2 + 4}\]

Step 6: Compose function expressions for (d)

For (d), substitute \(f(x)\) into \(g(x)\) to find \(g(f(x))\).Given \(f(x) = \sqrt{x+4}\) and \(g(x) = x^2\), substitute:\[g(f(x)) = g(\sqrt{x+4}) = (\sqrt{x+4})^2 = x + 4\]

Step 7: Multiplication of functions for (e)

For (e), multiply the functions \(f(t)\) and \(g(t)\).Given \(f(t) = \sqrt{t+4}\) and \(g(t) = t^2\), perform the multiplication:\[f(t) \, g(t) = \sqrt{t+4} \, t^2 = t^2 \sqrt{t+4}\]

Key Concepts

Composite Functions

Composite functions involve the process of combining two functions to create a new function. Think of it as a function inside another function, much like nesting dolls. If you have two functions, say \( f(x) \) and \( g(x) \), the composite function \( f(g(x)) \) means that you plug \( g(x) \) into \( f(x) \). This requires you to first evaluate \( g(x) \) and then use that result to find \( f(x) \).
In our example, to find \( f(g(x)) \), we substitute \( g(x) = x^2 \) into \( f(x) = \sqrt{x+4} \). Hence, \( f(g(x)) = \sqrt{(x^2)+4} \). It's important to follow the order of operations: inner function first, outer function second.

• Start with the inside function \( g(x) \).
• Substitute the result into the outer function \( f(x) \).
• Simplify the expression if possible.

This procedure is the essence of function composition and is a useful tool in various mathematical and applied contexts.

Function Evaluation

Function evaluation is the process of finding the value of a function for a given input. It's a fundamental concept in mathematics.
To evaluate a function like \( f(x) \) or \( g(x) \), you simply substitute the input value into the function's expression. Take note of our example functions: \( f(x) = \sqrt{x+4} \) and \( g(x) = x^2 \).
Suppose you want to evaluate these functions at \( x=1 \):

• For \( f(x) = \sqrt{x+4} \), substitute \( x = 1 \):
  \( f(1) = \sqrt{1+4} = \sqrt{5} \).
• For \( g(x) = x^2 \), substitute \( x = 1 \):
  \( g(1) = 1^2 = 1 \).

Understanding function evaluation is essential for working with more complex equations and for understanding how functions behave over their domains. It's the gateway to analyzing and applying functions effectively in many disciplines.

Multiplication of Functions

Multiplying functions is another fundamental operation where you take two functions and create a new function from them by multiplying their outputs for the same input.
For example, let's see what happens when we multiply the functions \( f(t) \) and \( g(t) \) given by \( f(t) = \sqrt{t+4} \) and \( g(t) = t^2 \):

• Multiply the functions directly:
  \( f(t) \cdot g(t) = \sqrt{t+4} \cdot t^2 = t^2\sqrt{t+4} \).

This new combination function \( f(t)g(t) \) tells us how the functions \( f(t) \) and \( g(t) \) interact. Multiplying functions can help in finding intersections, maximizing expressions, or combining data behaviors in practical applications.
Remember: when multiplying functions, always ensure that the operation is valid over the entire domain of interest.