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Chapter 3: Problem 322

In the following exercises, evaluate the function. $$ g(x)=4 x^{2}-3 x ; g(3) $$

Short Answer

Expert verified
g(3) = 27

Step by step solution

01

- Write down the function

The given function is: g(x) = 4x^2 - 3x
02

- Substitute the value

Substitute the value of x with 3 in the function: g(3) = 4(3)^2 - 3(3)
03

- Calculate the exponent

Calculate the exponent first: (3)^2 = 9, so g(3) = 4(9) - 3(3)
04

- Multiply the constants

Multiply the constants next: 4(9) = 36 and 3(3) = 9, so g(3) = 36 - 9
05

- Perform the final subtraction

Finally, subtract the second term from the first term: 36 - 9 = 27

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Key Concepts

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

Substitute Value
When evaluating a function, the first step is to substitute the given value of the variable into the function. For example, if you have a function \(g(x) = 4x^2 - 3x\) and you need to evaluate \(g(3)\), you replace \(x\) with 3. This changes the function into \(g(3) = 4(3)^2 - 3(3)\). Substituting the value correctly is crucial because any error at this stage will carry through the entire evaluation.
Calculate Exponent
After substituting the value into the function, the next step involves handling any exponents. In our example, once we have substituted \(x\) with 3, we have \(g(3) = 4(3)^2 - 3(3)\). Here, you need to calculate \((3)^2\).

An exponent indicates that you multiply the number by itself; hence, \((3)^2 = 3 * 3 = 9\). This step transforms our function into \(g(3) = 4(9) - 3(3)\).
Function Evaluation
Now that the exponent is calculated, you proceed to the multiplications in the function. From the previous step, we have \(g(3) = 4(9) - 3(3)\).

First, solve the multiplication:
  • \(4(9) = 36\)
  • \(3(3) = 9\)
Upon performing these multiplications, the function simplifies to \(g(3) = 36 - 9\).
Final Subtraction
The last step in function evaluation is to perform any remaining arithmetic operations. In our function, we have simplified everything to \(g(3) = 36 - 9\).

Now, subtract the second term from the first term:
  • \(36 - 9 = 27\)
Hence, the value of the function \(g(x)\) when \(x = 3\) is \(g(3) = 27\).

Always ensure each step is performed carefully, as mistakes in simpler operations can lead to incorrect answers.

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