Problem 73 \(\begin{array}{llll}\text { In ... [FREE SOLUTION]

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Chapter 5: Problem 73

\(\begin{array}{llll}\text { In each example, find } & (f+g)(x) & (b)(f+g)(2) & (c)(f-g)(x) & \text { (d) }(f-g)(-3) .\end{array}\) \(f(x)=2 x^{2}-4 x+1\) and \(g(x)=5 x^{2}+8 x+3\)

Short Answer

Expert verified

(f+g)(x) = 7x^2 + 4x + 4, (f+g)(2) = 40, (f-g)(x) = -3x^2 - 12x - 2, (f-g)(-3) = 7.

Step by step solution

Identify Given Functions

The given functions are:\(f(x)=2x^{2}-4x+1\)\(g(x)=5x^{2}+8x+3\)

Calculate \(f+g\)(x)

Combine the functions:\(f(x) + g(x) = (2x^{2}-4x+1) + (5x^{2}+8x+3) = 7x^{2}+4x+4\).

Evaluate \(f+g\)(2)

Substitute 2 into \(f+g\)(x):\(f+g(2) = 7(2)^{2}+4(2)+4 = 7(4)+8+4 = 28+8+4 = 40\).

Calculate \(f-g\)(x)

Combine the functions:\(f(x) - g(x) = (2x^{2}-4x+1) - (5x^{2}+8x+3) = -3x^{2}-12x-2\).

Evaluate \(f-g\)(-3)

Substitute -3 into \(f-g\)(x):\(f-g(-3) = -3(-3)^{2}-12(-3)-2 = -3(9)+36-2 = -27+36-2 = 7\).

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Key Concepts

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

Polynomial functions

Polynomial functions are an essential aspect of high school algebra and beyond. A polynomial function is an expression that involves variables raised to whole number exponents and coefficients. For example, the functions given in the exercise - \(f(x) = 2x^2 - 4x + 1\) and \(g(x) = 5x^2 + 8x + 3\) - are both polynomial functions.

Polynomials can have different degrees. The degree of a polynomial is the highest power of the variable within the expression. In our functions, both \(f(x)\) and \(g(x)\) are of degree 2 because the highest power of \(x\) is 2.

Here are some key characteristics of polynomial functions:

They can be added, subtracted, and multiplied.
They are continuous and smooth curves when graphed.
They can have multiple terms, constants, and variables.

Function addition

Function addition is the process of combining two functions to create a new function. In the provided exercise, we need to find \((f+g)(x)\), which means adding the functions \(f(x)\) and \(g(x)\).

Here鈥檚 how to do it:

Write down the functions to be added.
Combine like terms, meaning terms with the same power of \(x\).
Simplify the expression if needed.

For example:

\[f(x) = 2x^2 - 4x + 1\]
\[g(x) = 5x^2 + 8x + 3\]

Adding these functions, we get:

\((f + g)(x) = (2x^2 - 4x + 1) + (5x^2 + 8x + 3) = 7x^2 + 4x + 4\)

Function addition can be applied to evaluate at specific values too. For instance, to find \((f + g)(2)\), substitute \(2\) for \(x\) in the combined function:

\((f+g)(2) = 7(2)^2 + 4(2) + 4 = 28 + 8 + 4 = 40\)

Function subtraction

Function subtraction is similar to function addition, but instead of adding, you subtract the functions. In the exercise, we are asked to find \((f - g)(x)\). Here are the steps to subtract functions:

Write down the functions to be subtracted.
Subtract like terms, ensuring the correct signs are used.
Simplify the resulting expression.

For example:

\[f(x) = 2x^2 - 4x + 1\]
\[g(x) = 5x^2 + 8x + 3\]

Subtracting these functions, we get:

\((f - g)(x) = (2x^2 - 4x + 1) - (5x^2 + 8x + 3) = -3x^2 - 12x - 2\)

Just like with addition, function subtraction can be applied to specific values. To find \((f - g)(-3)\), substitute \(-3\) for \(x\) in the combined function:

\((f - g)(-3) = -3(-3)^2 - 12(-3) - 2 = -27 + 36 - 2 = 7\)

Function evaluation

Function evaluation involves finding the value of a function for a specific input. It is a fundamental skill in algebra and calculus. To evaluate a function, you simply substitute the given value for the variable \(x\) in the function's expression.

Let's break it down with examples from the exercise:

To evaluate \((f+g)(2)\), we first find \(f+g\) and then substitute \(2\) for \(x\).
To evaluate \((f-g)(-3)\), we first find \(f-g\) and then substitute \(-3\) for \(x\).

For instance:

\((f+g)(2) = 7(2)^2 + 4(2) + 4 = 28 + 8 + 4 = 40\)
\((f-g)(-3) = -3(-3)^2 - 12(-3) - 2 = 81 + 60 - 2 = 7\)

Remember these steps:

Identify the function and the value to be substituted.
Replace the variable \(x\) with the specific value.
Calculate the result step by step.

By following these steps, you can evaluate any function for any given input.

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91影视

\(\begin{array}{llll}\text { In each example, find } & (f+g)(x) & (b)(f+g)(2) & (c)(f-g)(x) & \text { (d) }(f-g)(-3) .\end{array}\) \(f(x)=2 x^{2}-4 x+1\) and \(g(x)=5 x^{2}+8 x+3\)

Short Answer

Step by step solution

Identify Given Functions

Calculate \(f+g\)(x)

Evaluate \(f+g\)(2)

Calculate \(f-g\)(x)

Evaluate \(f-g\)(-3)

Key Concepts

Polynomial functions

Function addition

Function subtraction

Function evaluation

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