Question

Given \(g(x)=x^{3}\) and \(f(x)=4-3 x,\) find $$f[g(3)]$$

Short answer

The value of f[g(3)] is -71.

Step-by-step solution

Evaluate the inner function

First, evaluate the inner function, which is g(x), at the given value. In this case, compute g(3).

Substitute the result into the outer function

After finding g(3), substitute this value into the outer function f(x) as the input.

Simplify the expression

Simplify the expression to find the final value of f[g(3)].

Key concepts

Function Composition

Understanding function composition is critical when dealing with multiple mathematical functions that operate in sequence. By definition, the composition of two functions, say f and g, is the process of applying one function to the results of another. This is notated as \( f(g(x)) \), which reads as 'f of g of x'. In the given exercise, you're asked to find \( f[g(3)] \), meaning you first evaluate \( g(x) \) at \( x = 3 \), and then apply the function \( f \) to the result of \( g(3) \).

It's like a two-step process. Imagine if \( g(x) \) was a machine that shapes clay, and \( f(x) \) was a paint station. If you wanted to get a clay model that's shaped and painted, you'd first pass it through the clay shaper (\( g \)), and then, with the shaped model in hand (the output from \( g \)), you'd move to the paint station (\( f \)). Each function depends on the output of the previous one to produce the final result.

Mathematical Functions

Mathematical functions are like special machines or rules that take an input and turn it into an output following a specific process. A function \( f \) that takes an input \( x \) is commonly written as \( f(x) \). This notation signifies that there's a relationship where for each input x, there is a single output, which results from applying the function's rule to x.

Think of functions as a cooking recipe: you start with ingredients (inputs), follow the cooking steps (function's rule), and end up with a dish (output). In our current example, the functions are \( g(x) = x^3 \) and \( f(x) = 4 - 3x \). These functions tell us exactly what to do with any input value of \( x \). For instance, if we input 3 into \( g(x) \), according to its 'recipe', we cube the 3 to get 27.

Function Evaluation

Function evaluation is the process by which we apply the rule of a function to a particular input value to produce an output. It's simply a matter of substituting the input value into the function and calculating the result. When you are asked to evaluate \( f(x) \) at \( x = a \), you replace every occurrence of \( x \) in the function with \( a \) and then calculate.

How do you go about this? First, identify the input value. Then, substitute that value into the function's equation wherever you see an \( x \). Perform any necessary calculations, and hey presto, you have your output! Using our example, to evaluate \( g(3) \), we replace \( x \) with 3 in \( g(x) = x^3 \), giving us \( g(3) = 3^3 = 27 \).

Simplifying Expressions

Simplifying expressions in mathematics means rewriting them in a more concise and manageable form without changing their value. The goal here is to make the expression easier to understand and work with. This might involve combining like terms, using the distributive property, or canceling out terms where possible.

In function composition, after we've evaluated the inner function and substituted into the outer function, we need to simplify to get our final answer. This step is crucial to avoid errors and to reveal the most reduced form of the result. In the provided example, after substituting \( g(3) \) into \( f \), we get \( f(27) = 4 - 3(27) \). Simplifying, we subtract 3 times 27 from 4, leading us to the solution \( f[g(3)] = -77 \). This simplification process is like tidying up after cooking; cleaning the counters and putting away dishes gives you a clear space and a ready-to-serve meal.