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Chapter 0: Problem 53

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(-1)+g(-1) $$

Short Answer

Expert verified
-2

Step by step solution

01

- Evaluate f(-1)

First, substitute -1 into the function f(x). f(-1)=3(-1)^2-(-1)=3(1)+1=3+1=4
02

- Evaluate g(-1)

Next, substitute -1 into the function g(x). g(-1)=4(-1)-2=-4-2=-6
03

- Add the results

Finally, add the results from Step 1 and Step 2. f(-1) + g(-1) = 4 + (-6) = -2

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Key Concepts

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

Function Evaluation
When evaluating a function, you are essentially plugging in a specific value of the variable (usually denoted as x) into the function to calculate its output. In the given problem, we are asked to find the values of two functions, f(x) and g(x), at the point x = -1.
Here鈥檚 how you generally evaluate a function at a specific point:
  • Identify the function and the value of x to be substituted.
  • Replace every instance of x in the function with the given value.
  • Perform the arithmetic operations to find the result.
Let鈥檚 break down the process using the given functions:
  • For the quadratic function, find f(-1) by substituting -1 into f(x).
  • Similarly, for the linear function g(x), find g(-1) by substituting -1 into g(x).
Quadratic Function
A quadratic function is a type of polynomial function with the form:
\( f(x) = ax^2 + bx + c \)
In our problem, the quadratic function is given by:
\( f(x) = 3x^2 - x \)
To evaluate this function at x = -1, follow these steps:
  • Substitute -1 for x in the equation, making it f(-1) = 3(-1)^2 - (-1).
  • Simplify the equation inside the parentheses: (-1)^2 = 1.
  • Multiply the constants: 3 * 1 = 3.
  • Simplify further: 3 + 1 = 4.
So, \( f(-1) = 4 \). This means that when x = -1, the output of function f(x) is 4.
Linear Function
A linear function is a function of the form:
\( g(x) = mx + b \)
where m and b are constants. This represents a straight line graph. In our problem, the linear function is given by:
\( g(x) = 4x - 2 \)
To evaluate this function at x = -1, follow these steps:
  • Substitute -1 for x in the equation, making it g(-1) = 4(-1) - 2.
  • Multiply the constants: 4 * (-1) = -4.
  • Simplify further: -4 - 2 = -6.
So, \( g(-1) = -6 \). This means that when x = -1, the output of function g(x) is -6.
Finally, add the values of f(-1) and g(-1):
  • f(-1) + g(-1) = 4 + (-6) = -2.
Therefore, the final answer to the problem is -2.

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