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Chapter 9: Problem 46

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (f+h)(-2) $$

Short Answer

Expert verified
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Step by step solution

01

Define Functions

Given the functions: 1. \( f(x) = x^2 - 9 \) 2. \( g(x) = 2x \) 3. \( h(x) = x - 3 \)
02

Find \( f(x) + h(x) \)

Combine the functions \( f(x) \) and \( h(x) \): \[ f(x) + h(x) = (x^2 - 9) + (x - 3) \] Simplify the expression: \[ f(x) + h(x) = x^2 + x - 12 \]
03

Evaluate at \( x = -2 \)

Substitute \( x = -2 \) into the expression: \[ f(-2) + h(-2) = (-2)^2 + (-2) - 12 \] Calculate the components step by step: \[ (-2)^2 = 4 \] \[ 4 + (-2) = 2 \] \[ 2 - 12 = -10 \]

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Key Concepts

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

Combining Functions
Combining functions is a fundamental concept in algebra. This process involves adding, subtracting, multiplying, or dividing two or more functions to create a new function. In this problem, we combined the functions \(f(x)\) and \(h(x)\).

To combine the functions \(f(x)\) and \(h(x)\), we simply add the corresponding expressions. Given:
  • \(f(x) = x^2 - 9\)
  • \(h(x) = x - 3\)
Combining them, we get:

\[ f(x) + h(x) = (x^2 - 9) + (x - 3) \]

After combining the functions, our new expression is:

\[ f(x) + h(x) = x^2 + x - 12 \]

It's important to correctly perform operations on each corresponding part to ensure there are no mistakes.
Evaluating Functions
Evaluating functions means finding the value of the function for a particular input. In this exercise, we evaluate the combined function \(f(x) + h(x)\) at \(x = -2\).

Once we have our combined function from the previous step, which is:

\[ f(x) + h(x) = x^2 + x - 12 \]

We substitute \(x = -2\) into this expression. This is how you do it step-by-step:
  • Substitute \(x = -2\) into the function: \[ f(-2) + h(-2) = (-2)^2 + (-2) - 12 \]
  • First, calculate the square: \[ (-2)^2 = 4 \]
  • Then add \(-2\): \[ 4 + (-2) = 2 \]
  • Finally, subtract 12: \[ 2 - 12 = -10 \]
So, \( (f+h)(-2) = -10 \). Evaluating functions often involves substituting values and simplifying expressions in a logical sequence.
Substitution Method
The substitution method is used frequently in algebra to replace variables with specific values or expressions. This method helps simplify solving, especially when dealing with combined functions or systems of equations.

In this particular problem, we are using substitution to evaluate the combined function. Here's a closer look at the steps:
  • Start with your combined function: \[ f(x) + h(x) = x^2 + x - 12 \]
  • Identify the value to be substituted, which is \(x = -2\).
  • Substitute \(-2\) for \(x\) into the combined function: \[ (-2)^2 + (-2) - 12 \]
  • Simplify each component of the expression to find the result.
Using substitution is handy because it allows us to handle complex expressions step-by-step. It breaks down the task and reduces the chance for errors, ensuring accurate solutions.

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Most popular questions from this chapter

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{h}{g}\right)\left(-\frac{3}{2}\right) $$

The perimeter \(x\) of a square with sides of length \(s\) is given by the formula \(x=4 s\) (a) Solve for \(s\) in terms of \(x\). (b) If \(y\) represents the area of this square, write \(y\) as a function of the perimeter \(x\). (c) Use the composite function of part (b) to find the area of a square with perimeter 6 .

For each pair of functions, find \(\left(\frac{f}{g}\right)(x)\) and give any \(x\) -values that are not in the domain of the quotient function. $$ f(x)=10 x^{2}-2 x, \quad g(x)=2 x $$

For each pair of functions, find \(\left(\frac{f}{g}\right)(x)\) and give any \(x\) -values that are not in the domain of the quotient function. $$ f(x)=27 x^{3}+64, \quad g(x)=3 x+4 $$

Solve each problem. The maximum load that a cylindrical column with a circular cross section can hold varies directly as the fourth power of the diameter of the cross section and inversely as the square of the height. A \(9-\mathrm{m}\) column \(1 \mathrm{~m}\) in diameter will support 8 metric tons. How many metric tons can be supported by a column \(12 \mathrm{~m}\) high and \(\frac{2}{3} \mathrm{~m}\) in diameter?
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