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Chapter 0: Problem 55

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(-5)-k(-5) $$

Short Answer

Expert verified
78

Step by step solution

01

- Find f(-5)

Substitute x = -5 into the function f(x) = 3x^2 - x. Calculate f(-5). f(-5) = 3(-5)^2 - (-5) = 3(25) + 5 = 75 + 5 = 80.
02

- Find k(-5)

Substitute x = -5 into the function k(x) = |x + 3|. Calculate k(-5). k(-5) = |-5 + 3| = |-2| = 2.
03

- Subtract k(-5) from f(-5)

Subtract the value found in Step 2 from the value found in Step 1: f(-5) - k(-5) = 80 - 2 = 78.

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Key Concepts

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

Function Evaluation
Understanding function evaluation is crucial in mathematics. A function is like a machine: you input a value (the input, or 'x'), and the function processes it to give an output. The notation for evaluating a function is usually written as \( f(x) = \text{expression} \), where 'f' indicates the function, 'x' is the input, and 'expression' shows how the input is processed. For example, if \( f(x) = 3x^2 - x \), evaluating \( f(-5) \) means substituting -5 for every x in the expression. This gives \( f(-5) = 3(-5)^2 - (-5) \), which simplifies to 80.
Absolute Value
The absolute value is a special function denoted by two vertical bars: \(|x|\). It measures the distance of a number from zero on the number line and is always non-negative. For example, \(|-5| = 5\) and \(|3| = 3\). In the exercise, the function \( k(x) = |x + 3| \) needs us to substitute \( x = -5 \), giving us \( k(-5) = |-5 + 3| = |-2| = 2 \). Absolute value strips the negative sign but leaves positive values unchanged.
Quadratic Function
Quadratic functions are polynomial functions of degree 2, generally written as \( f(x) = ax^2 + bx + c \). They graph as parabolas, which can either open upwards or downwards depending on the leading coefficient 'a'. In \( f(x) = 3x^2 - x \), we see the quadratic term \( 3x^2 \) makes it a quadratic function. Evaluating it for any input is straightforward by substituting the input value for x. For instance, \( f(-5) = 3(-5)^2 - (-5) = 80 \).
Step-By-Step Solutions
Breaking down problems into step-by-step solutions helps in understanding complex exercises. In the given problem, the first step is to evaluate \( f(-5) \), yielding 80. The next step involves finding \( k(-5) \), which is 2. Finally, we subtract the result of \( k(-5) \) from \( f(-5) \), resulting in 78. Following each step ensures clarity and avoids confusion, making problem-solving manageable. Simplifying each part of an equation and methodically solving them leads to the correct answer. This strategy is vital for mastering mathematical concepts.

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