Problem 94 - Precalculus Essentials

Question

Let \(f(x)=x^{2}-x+4\) and \(g(x)=3 x-5\) Find \(g(-1)\) and \(f(g(-1))\)

Step-by-Step Solution

Answer

\(g(-1) = -8\) and \(f(g(-1)) = 76\).

Step 1: Calculate \(g(-1)\)

Substitute -1 for 'x' in the function \(g(x)\). So, \(g(-1) = 3(-1) - 5 = -3 - 5 = -8\).

Step 2: Compute \(f(g(-1))\)

Now that we know that \(g(-1) = -8\), we substitute -8 for 'x' in the function \(f(x)\). So, \(f(g(-1)) = f(-8) = (-8)^{2} - (-8) + 4 = 64 + 8 + 4 = 76.\)

Key Concepts

Function Composition

In mathematics, function composition is a powerful operation that allows us to combine two or more functions to create a new function. It involves applying one function to the results of another. We denote the composition of two functions, f and g, as \( f(g(x)) \). This means you first evaluate g for a certain value of x, and then use the result as the input for the function f. It's like a multi-step journey, where x first goes through g, and then the output of g goes through f.

Understanding function composition is crucial because it helps to simplify complex problems and can reveal deeper insights about the functions involved. For the given exercise, we first find the value of \( g(x) \) when x equals -1, and then use this value to find the composite function \( f(g(-1)) \). This method is central to understanding how one function can influence the output of another, creating a new relation.

Function Evaluation

Function evaluation is the process of finding the output of a function for a particular input value. Essentially, it's answering the question, 'If I put this value into the function, what value will I get out?' The function acts like a machine that gives a certain output for a given input, according to its rule, which is the mathematical expression that defines the function.

For our example, evaluating the function \( g(x) \) when x is -1 means substituting -1 into every occurrence of x in the expression for g. We calculate this to get a numerical output. Then, evaluating the function \( f(x) \) with \( g(-1) \) as input follows the same substitution principle.

Polynomial Functions

Polynomial functions are some of the most common and widely used functions in algebra, which are formed by adding multiples of powers of the variable. A general polynomial function looks like \( a_{n}x^n + a_{n-1}x^{n-1} + ... + a_{2}x^2 + a_{1}x + a_{0} \), where the \( a_{i} \) are constants, and n is a non-negative integer that denotes the degree of the polynomial, which is the highest power of x present.

The function \( f(x) = x^2 - x + 4 \) in our example is a polynomial of degree 2, commonly known as a quadratic function. These functions are particularly interesting because they can model various phenomena, like projectile motion. In algebra, they help students learn important concepts such as function behavior and curve sketching.

Substitution Method

The substitution method is a vital algebraic technique used to simplify expressions or solve equations. It involves replacing a variable with another expression or a value. You 'substitute' a known value into an equation where that variable appears. This process reduces complexity and helps isolate variables for solving.

In our exercise, we use substitution twice. First, we substitute x with -1 in \( g(x) \), and then we substitute x with the result \( g(-1) \) into the function \( f(x) \). This step-by-step approach allows us to find out what happens when the input is not simply x, but another function of x. It's like solving a puzzle, and substitution is one of the key moves to unlock the final answer.