Problem 93 - College Algebra 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

The value of \(g(1) = -2\) and \(f(g(1)) = 10\)

Step 1: Compute \(g(1)\)

The function \(g(x) = 3x - 5\). Let's substitute \(x = 1\) in the equation. Thus, \(g(1) = 3(1) - 5 = -2\)

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

Now that we have \(g(1) = -2\), let's substitute \(x = -2\) in the function \(f(x) = x^2 - x + 4\). Thus, \(f(g(1)) = f(-2) = (-2)^2 - (-2) + 4 = 4 + 2 + 4 = 10\)

Key Concepts

Function Composition

Understanding function composition is crucial to mastering algebra. It refers to applying one function to the results of another. When composing functions, you take the output from one function and use it as the input for another. In algebraic terms, if you have two functions, say f and g, the composite function f(g(x)) means you first evaluate g(x) and then use that result as the input for f.

In the given exercise, g(1) is first computed and then used as the input for f. This illustrates function composition where g is composed with f, denoted as f(g(x)). It's like a sequence where the output of one function becomes the next one's input. Always remember to process functions from the inside out—evaluating the innermost function first.

Evaluating Functions

Evaluating functions is a basic but essential skill in algebra. This process involves finding the output of a function given a particular input. When asked to evaluate a function like g(x) = 3x - 5 at x = 1, you replace the variable x with 1 in the equation and simplify. For g(1), you calculate 3(1) - 5 to get -2.

It's important to follow the correct order of operations—parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right), often abbreviated as PEMDAS. After evaluating g(1), we turn to evaluate f(g(1)), which involves plugging the output from g into the function f. This step-by-step approach ensures clarity and minimizes errors in more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and represent mathematical ideas using numbers, variables, and operations. The expressions f(x) = x^2 - x + 4 and g(x) = 3x - 5 from the exercise are examples. They can be simplified, factored, or used within functions to evaluate specific values, like g(1) or f(g(1)).

Understanding how to manipulate these expressions allows you to solve equations, model real-world situations, and understand the behavior of functions. When working with algebraic expressions, always pay attention to coefficients, like the 3 in 3x, and operations. Breaking expressions down into smaller parts can help manage their complexity, especially when evaluating composite functions.