/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Find each function value. $$ ... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 63

Find each function value. $$ f(x)=x^{2}+4 x-3 . \text { Find } f(2) $$

Short Answer

Expert verified
The value of \(f(2)\) is 9.

Step by step solution

01

Identify the Function

The given function is \(f(x) = x^2 + 4x - 3\).
02

Substitute the Given Value

Substitute \(x = 2\) into the function \(f(x)\). This gives us \(f(2) = 2^2 + 4(2) - 3\).
03

Simplify the Expression

Calculate each term in the expression: \(2^2 = 4\), \(4 \times 2 = 8\), and \(-3\). Thus, \(f(2) = 4 + 8 - 3\).
04

Combine Like Terms

Add the terms together: \(4 + 8 - 3 = 9\). Therefore, \(f(2) = 9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitute values in functions
Substituting values in functions is the first step in evaluating a function. Imagine a function as a machine where you input a value and it provides an output based on a specific formula. In the given exercise, we have the function \( f(x) = x^2 + 4x - 3 \). To find \( f(2) \), we need to replace every occurrence of \( x \) in the function with 2. This substitution gives us: \( f(2) = 2^2 + 4(2) - 3 \). This step is crucial because without substituting the correct value, the function cannot be evaluated correctly. Always ensure you carefully replace the variable with the given value to avoid mistakes.
Simplify expressions
Simplifying expressions involves performing arithmetic operations to reduce the expression to its simplest form. Once you have substituted the values in the function, you need to simplify the resulting expression. For our example, we substitute \( x = 2 \) into the function to get \( 2^2 + 4(2) - 3 \). Here's how to simplify it:
  • Calculate \( 2^2 \): this is 4.
  • Next, calculate \( 4 \times 2 \): this gives us 8.
  • The only remaining term is -3.
Put all these together by following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction - also known as PEMDAS). Now, our expression is simplified to: \( 4 + 8 - 3 \).
Find function values
The final step in evaluating a function is to combine the terms to find the function value. This involves adding or subtracting the simplified terms to get the final result. From the previous section, we simplified the expression to \( 4 + 8 - 3 \). Now, let's find the function value:
  • Add 4 and 8 to get 12.
  • Then subtract 3 from 12 to get 9.
So, after combining the terms: \( f(2) = 9 \). Therefore, the value of the function \( f(x) \) when \( x \) is 2 is 9. This step completes the process of function evaluation, which is critical for understanding the behavior of functions for different input values.

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