Problem 13 - Technical Mathematics with Calculus

Question

Substitute the literal values into each function. $$\text { If } f(x)=2 x^{2}+4, \text { find } f(a)$$.

Step-by-Step Solution

Answer

The function after substitution is \(f(a) = 2a^2 + 4\).

Step 1: Identify the function and variable

First, we'll take the given function, which is \(f(x) = 2x^2 + 4\). Next, we identify the variable that we need to substitute into the function, which in this case is \(a\).

Step 2: Substitute the variable 'a' into the function

Now, substitute \(a\) for each instance of \(x\) in the function to find \(f(a)\). So, the new function will look like \(f(a) = 2a^2 + 4\).

Step 3: Simplify the function, if necessary

No further simplification is required in this case since no specific value of \(a\) is given. We leave the function as \(f(a) = 2a^2 + 4\).

Key Concepts

Function Evaluation

Function evaluation is like using a very specific recipe where your ingredients are the variables, and the outcome is your final dish. In algebra, a function represents a rule that takes certain inputs and turns them into outputs. When we’re given a function, such as, and asked to find , we're essentially being asked to prepare the same dish, but this time with a different ingredient, which is the variable .

To do this correctly, we substitute the variable wherever we see in the original function, and this is how we 'evaluate' the function for a new variable. It's important here not to change any other part of the function. Think of it as ensuring that you're only changing the flavor but not the method of cooking. By substituting correctly and not altering the rest of the 'recipe,' we stay true to the function's original intention.

Algebraic Expressions

Algebraic expressions are a mix of numbers, variables, and operations (like addition and multiplication) that represent a particular quantity. Think of an algebraic expression as a code that needs to be deciphered or a sentence that is missing some words, and you need to fill in the blanks to make sense of it.

An understanding of algebraic expressions is crucial when working with functions. They are like a set of instructions that tell you how to combine the ingredients - our variables - to get your final outcome. For instance, in the expression <2x^2 + 4, the instruction is to square the variable , multiply it by 2, and then add 4. Here, 'instrument' represents an incomplete sentence until you substitute a specific value in place of .

Simplifying Expressions

Simplifying expressions in algebra is akin to cleaning up after you've followed a recipe. It's about making the end result - the expression - as neat as possible. This usually involves combining like terms, reducing fractions, or factoring.

In our case, is already quite tidy. There are no like terms to combine, so no further simplifying is needed. However, had we been given numeric value for , we might perform computations to simplify. For instance, if , then . Simplification often helps in understanding what the expression will look like numerically and makes it easier to handle, especially in more complex problems.