Problem 125 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((f g)\left(-\frac{4}{7}\right)\)

Step-by-Step Solution

Answer

So, \((f g)\left(-\frac{4}{7}\right) = -\frac{512}{49}\).

Step 1: Evaluate the functions at the given point

First, we need to find the values of \(f(x)\) and \(g(x)\) at \(x = -\frac{4}{7}\). \n For \(f(x)\), we substitute \(x = -\frac{4}{7}\) into the equation to obtain: \n \(f(-\frac{4}{7}) = 14(-\frac{4}{7}) - 3 = -8.\) \n For \(g(x)\), we substitute \(x = -\frac{4}{7}\) into the equation to get: \n \(g(-\frac{4}{7}) = 8(-\frac{4}{7})^2 = \frac{64}{49}. \)

Step 2: Calculation of the product of functions

The product of the functions at \(x = -\frac{4}{7}\) is given by: \( (f g)\left(-\frac{4}{7}\right) = f\left(-\frac{4}{7}\right) \cdot g\left(-\frac{4}{7}\right) = -8 \cdot \frac{64}{49}\).

Step 3: Simplifies the expression

Now, finally simplifying the expression gives: \( -8 \cdot \frac{64}{49} = -\frac{512}{49}\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics that involves finding the output value of a function for a given input value. It's like plugging in a specific number into a formula and seeing what number comes out. In this case, we work with two functions, \(f(x) = 14x - 3\) and \(g(x) = 8x^2\), and we are asked to evaluate these functions at \(x = -\frac{4}{7}\).

For \(f(x)\), we substitute the given \(x\)-value directly into \(f\) to find \(f(-\frac{4}{7})\). We compute this by following the algebraic rules, such as multiplication and addition. Similarly, for \(g(x)\), we place \(x = -\frac{4}{7}\) into \(g\) and simplify, which involves squaring the number and then multiplying by 8. Function evaluation is essential because it allows us to understand how a function behaves and what values it takes for different inputs.

Composition of Functions

When we talk about the composition of functions, we're referring to the combining of two functions where the output of one function becomes the input to another. It's like feeding the output from one machine into another and seeing what comes out the other end! However, in the exercise, we are actually not asked for a composition of functions but for the product of the two functions evaluated at the same point.

To clarify, composition would have been \(f(g(x))\) or \(g(f(x))\) which is not what we're asked to find. Instead, we compute the product, which involves multiplying the outputs of both \(f\) and \(g\) when they are individually evaluated at \(x = -\frac{4}{7}\). This concept is key in understanding how different functions can operate together, either by being composed or multiplied.

Algebraic Manipulation

Algebraic manipulation involves the skills and techniques used to transform mathematical expressions into different, often simpler, forms. The purpose is to make it easier to understand or to find a solution to an equation or inequality. During the process of solving our given problem, we utilize several algebraic techniques.

We start by substituting values, followed by operating according to the order of operations (PEMDAS/BODMAS), and finally simplifying fractions. When we multiply \(f(-\frac{4}{7})\) by \(g(-\frac{4}{7})\), we're performing an algebraic manipulation to find the product. The last step is simplification, where we take the product of the two numbers and reduce it if possible. In our case, the multiplication led to \( -8 \cdot \frac{64}{49} \), which can’t be simplified further, so the final answer remains \( -\frac{512}{49}\). Algebraic manipulation is a crucial skill in all areas of mathematics, allowing one to streamline expressions and solve equations effectively.