Problem 8 - Thomas Calculus

Question

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=3 x+4, \quad g(x)=2 x-1, \quad h(x)=x^{2} $$

Step-by-Step Solution

Answer

The formula for \(f \circ g \circ h\) is \(6x^2 + 1\).

Step 1: Understand the Composition

The problem asks for the composition of three functions, which is written as \((f \circ g \circ h)(x)\). This means we need to find \(f(g(h(x)))\). This will involve applying the innermost function \(h(x)\), followed by \(g\), then applying \(f\) to the result.

Step 2: Evaluate the Innermost Function

Start with the innermost function, \(h(x) = x^2\). This means for any input \(x\), the output of \(h\) is \(x^2\).

Step 3: Apply the Second Function

With \(h(x) = x^2\), we need to apply \(g\) to this result: \(g(h(x)) = g(x^2)\). Substitute \(x^2\) into \(g(x) = 2x - 1\), getting \(g(x^2) = 2(x^2) - 1 = 2x^2 - 1\).

Step 4: Apply the Outermost Function

Now we apply \(f\) to the result of \(g(h(x))\). Substitute \(2x^2 - 1\) into \(f(x) = 3x + 4\), giving \(f(2x^2 - 1) = 3(2x^2 - 1) + 4\).

Step 5: Simplify the Expression

Simplify the expression \(f(2x^2 - 1) = 3(2x^2 - 1) + 4\). Calculate \(3(2x^2 - 1) = 6x^2 - 3\), then add 4: \((6x^2 - 3) + 4 = 6x^2 + 1\). Thus, \((f \circ g \circ h)(x) = 6x^2 + 1\).

Key Concepts

composite functions

In mathematics, composite functions are an essential concept that deals with combining two or more functions. The notation for composing functions is typically written as \( (f \circ g \circ h)(x) \), which means you first apply the innermost function, followed by the next function, and finally the outermost function.
For the given problem, we have three functions: \( f(x) = 3x + 4 \), \( g(x) = 2x - 1 \), and \( h(x) = x^2 \). To compose these functions, we follow the sequence:

• Start with the innermost function \( h(x) \), which will transform \( x \) into \( x^2 \).
• Apply the result from \( h(x) \) to \( g \), thus getting \( g(x^2) \).
• Finally, apply \( f \) to the outcome from \( g(h(x)) \), getting \( f(g(h(x))) \).

By understanding this straightforward process, you’ll see how composite functions can create complex transformations from simple operations. Remember, the order in which you compose functions can greatly impact the result.

algebraic manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to make them easier to work with or to solve equations. When dealing with composite functions, you'll often need these skills to move from one step to the next effectively.
In our problem, once we apply function \( h(x) = x^2 \), we need to insert this result into function \( g(x) = 2x - 1 \). This requires replacing each instance of \( x \) in \( g(x) \) with \( x^2 \), which leads to \( g(x^2) = 2(x^2) - 1 = 2x^2 - 1 \).

• Substitute expressions into each other carefully, always double-checking your work.
• Simplify expressions by combining like terms or using distributive properties as necessary.

Finally, this result \( 2x^2 - 1 \) is substituted into \( f(x) = 3x + 4 \). Applying algebraic manipulation again, we get \( f(2x^2 - 1) = 3(2x^2 - 1) + 4 \). Using distributive properties simplifies to \( 6x^2 - 3 + 4 \), which further combines to \( 6x^2 + 1 \).
Through algebraic manipulation, the solution becomes refined and clearly represented.

mathematical functions

Mathematical functions serve as the foundation for many concepts in algebra and calculus. A function is essentially a rule that assigns each input exactly one output. Understanding how functions work is crucial for composing functions and manipulating them algebraically.
In the discussed exercise, we are working with three different functions:

• Linear Function: \( f(x) = 3x + 4 \). This provides a straight line graph with a slope of 3 and a y-intercept of 4.
• Linear Function: \( g(x) = 2x - 1 \). Also a straight line, with a slope of 2 and a y-intercept of -1.
• Quadratic Function: \( h(x) = x^2 \). This results in a parabolic graph that opens upwards.

Each of these functions transforms inputs in unique ways, which means that combining them through composition can lead to varied and complex outputs.
When tackling composite functions, always keep in mind the fundamental properties of the functions you are dealing with, as these affect the final composed function. Pay special attention to how each function's input is transformed into an output, which serves as the input for the next function in the composition.