Problem 43 - Algebra 2

Question

If \(f(x)=4 x, g(x)=2 x-1,\) and \(h(x)=x^{2}+1,\) find each value. $$ [f \circ(h \circ g)](3) $$

Step-by-Step Solution

Answer

The value of \([f \circ (h \circ g)](3)\) is 104.

Step 1: Understand the Composition of Functions

In this exercise, we need to find \[[f \circ (h \circ g)](3)onumber\]. This means we are first finding the value of \((h \circ g)(3)\) and then applying \(f\). The composition \((f \circ g)(x)\) is equivalent to \(f(g(x))\).

Step 2: Find \(g(3)\)

To begin solving \((h \circ g)(3)\), we first evaluate \(g(x)\) at \(x = 3\). Given:\[g(x) = 2x - 1\]So,\[g(3) = 2(3) - 1 = 6 - 1 = 5\].Now, we know that \(g(3) = 5\).

Step 3: Evaluate \(h(g(3))\)

Next, substitute \(g(3) = 5\) into \(h(x)\) to find \(h(g(3))\). Given:\[h(x) = x^2 + 1\]So,\[h(5) = 5^2 + 1 = 25 + 1 = 26\].Therefore, \((h \circ g)(3) = 26\).

Step 4: Compute \(f(h(g(3)))\)

Finally, substitute the result \((h \circ g)(3) = 26\) into \(f(x)\) to find \(f(h(g(3)))\). Given:\[f(x) = 4x\]So,\[f(26) = 4 \times 26 = 104\].Therefore, \([f \circ (h \circ g)](3) = 104\).

Key Concepts

Algebraic Functions

Algebraic functions are expressions composed of variables and constants using fundamental arithmetic operations—addition, subtraction, multiplication, and division. They often include exponents, such as in polynomial functions. These expressions are foundational in algebra and are used extensively in calculus and other advanced mathematics fields.
In this context, functions like \(f(x) = 4x\), \(g(x) = 2x - 1\), and \(h(x) = x^2 + 1\) are algebraic because they involve simple arithmetic with variables and constants. Each algebraic function takes an input, performs a specified operation, and provides an output.
By understanding algebraic functions, we can solve various mathematical problems, such as compositions, by substituting expressions into each function to determine their effect.

Evaluation of Functions

The evaluation of functions involves substituting a specific value for the variable, typically denoted as \(x\), and then performing the indicated operations to find the output.
In our exercise, the task involves evaluating three separate functions— \(f(x)\), \(g(x)\), and \(h(x)\). This process is central to function composition as it requires an intermediate step where values are substituted and results recalculated.

• Start by evaluating the innermost function \(g(x)\) using the given value \(x = 3\), to find \(g(3) = 5\).
• Next, use this result as the input for the next function \(h(x)\), giving \(h(g(3)) = h(5) = 26\).
• Finally, substitute this into \(f(x)\) to obtain \(f(26) = 104\).

Function evaluation is crucial for understanding how changes in input affect the output.

Mathematical Operations

Mathematical operations in function composition act as a way to transform and manipulate algebraic expressions. These operations include adding, subtracting, multiplying, and applying powers to variables or numbers.
In our problem, composition involves a specific order of operations: first evaluate \(g(x)\), then apply \(h(x)\), and finally use \(f(x)\). Each step simplifies the expression until the final result is achieved.

• When evaluating \(g(x) = 2x - 1\), multiplication and subtraction occur.
• For \(h(x) = x^2 + 1\), exponentiation and addition take place.
• The function \(f(x) = 4x\) involves a simple multiplication.

Understanding each of these operations allows us to effectively solve compositions like \([f \circ (h \circ g)](3)\), leading to the final answer of 104.