Problem 34 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of \(f[g(-1)]\) is \(-12\).

Step 1: Find g(-1)

To solve for \(g(-1)\), substitute \(-1\) into the function \(g(x)\). The function is \(g(x) = 2x - 1\). Now, calculate: \[ g(-1) = 2(-1) - 1 = -2 - 1 = -3 \] Thus, \(g(-1) = -3\).

Step 2: Substitute the result of g(-1) into f

Now that we have \(g(-1) = -3\), we need to find \(f(g(-1)) = f(-3)\). The function \(f(x)\) is given by \(f(x) = 4x\). Substitute \(-3\) into \(f(x)\):\[ f(-3) = 4(-3) = -12 \] Hence, \(f[g(-1)] = f(-3) = -12\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept used to understand how functions behave when applied to specific input values. Evaluating a function means replacing the variable in the function's expression with an actual number.
This process reveals the function's output for that particular input.
To evaluate a function:

• Identify the given function and its expression, such as \(f(x) = 4x\).
• Substitute the given input value into the expression, replacing every occurrence of the variable with this input.
• Perform the arithmetic operations indicated by the function expression.

In our case, when we compute \(f(-3)\), it involves substituting \(-3\) into \(f(x) = 4x\), leading to a result of -12.
This highlights the critical role of substitution in function evaluation, allowing us to determine specific outputs from various input values.

Algebraic Functions

Algebraic functions are functions constructed from algebraic operations like addition, subtraction, multiplication, division, and exponentials.
They form the backbone of many mathematical models and applications.
An algebraic function like \(g(x)=2x-1\) involves simple operations:

• Addition/Subtraction: adjusting the function's output by constant amounts.
• Multiplication/Division: scaling the function's input, often changing the shape or slope of its graph.

When dealing with an algebraic function, you use these operations to directly modify the input value and calculate the output.
In solving our original problem, \(g(x)=2x-1\) is evaluated at \(x=-1\), performing multiplication and subtraction to yield \(g(-1) = -3\).
Understanding the role of each operation helps in predicting how changes in input will affect the output.

Nested Functions

Nested functions involve the use of one function inside another, meaning the output of one function becomes the input of another.
This is common in complex mathematical computations and provides a versatile tool for problem-solving.
To solve nested functions, follow these steps:

• Identify all functions involved, such as \(f(x)\) and \(g(x)\).
• Evaluate the innermost function first, using its input to find its output.
• Use the result as the input for the outer function.

In the example \(f[g(-1)]\), \(g(-1)\) is computed to be -3, which then becomes the input for \(f(x)=4x\).
By evaluating \(f(-3)\) using the result from \(g(x)\), we simplify the nested operation to find the final value, -12.
This illustrates how function composition, through nesting, requires careful step-by-step evaluation to achieve the correct result.