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Chapter 2: Problem 43

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find and simplify each of the following. $$g(-2)$$

Short Answer

Expert verified
The value of \(g(-2)\) is \(-11\).

Step by step solution

01

Identify the function

The function to be used is \(g(x) = -x^2 + 4x + 1\).
02

Substitute \(x = -2\)

Replace \(x\) in the function \(g(x)\) with \(-2\). Thus, \(g(-2) = -(-2)^2 + 4(-2) + 1\).
03

Calculate \((-2)^2\)

Compute the square of \(-2\), which is \((-2)^2 = 4\).
04

Substitute and simplify

Substitute \(4\) back into the equation. \(g(-2) = -4 + 4(-2) + 1\).
05

Calculate \(4(-2)\)

Compute the product of \(4\) and \(-2\), which is \(4 \times -2 = -8\).
06

Combine terms

Combine the terms in the equation: \(g(-2) = -4 - 8 + 1\).
07

Simplify the expression

Simplify the combined terms: \(-4 - 8 + 1 = -11\).

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Key Concepts

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

substitution
Substitution is a powerful and essential mathematical process where we replace variables in an expression with their given values. In our exercise, we started with the function $$g(x)=-x^2+4x+1.$$ We needed to find the value of this function when $$x=-2.$$ This means we substitute -2 for every x in the equation.
So, the substitution looks like this: $$g(-2) = -(-2)^2 + 4(-2) + 1.$$ By performing substitution correctly, we can then focus on simplifying the resulting expression. Remember, substitution is widely used in solving equations, evaluating functions, and more, making it crucial for your math toolkit.
simplifying expressions
Simplifying expressions helps in reducing them to their most basic form, making them easier to work with. After substituting -2 into the function, the equation becomes: $$-(-2)^2 + 4(-2) + 1.$$ To simplify this, follow these steps:
  • First, calculate the square of -2: $$(-2)^2 = 4.$$
  • Then, replace this back into the equation: $$g(-2) = -4 + 4(-2) + 1.$$
  • Next, calculate $$4(-2) = -8$$ and substitute it into the expression: $$g(-2) = -4 - 8 + 1.$$
  • Lastly, combine the constants: ewline $$-4 - 8 = -12$$ ewline Then, $$-12 + 1 = -11.$$
Simplifying expressions step-by-step ensures accuracy and makes complex equations more manageable.
quadratic function
Quadratic functions are polynomial functions of degree 2, generally expressed in the form $$ax^2 + bx + c.$$ In our exercise, the quadratic function is $$g(x)=-x^2+4x+1.$$ These functions plot as parabolas on a graph and can open upwards or downwards based on the coefficient of $$x^2.$$
Key characteristics of quadratic functions include:
  • The vertex, representing the maximum or minimum point of the parabola.
  • The axis of symmetry, a vertical line passing through the vertex.
  • The roots or zeros, which are the x-intercepts of the function where $$g(x)=0$$.

Understanding quadratic functions is foundational for solving and graphing various mathematical problems. Whether you're finding evaluation values, like in our example, or determining roots using the quadratic formula, mastering these functions is essential for success in algebra and higher-level math.

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Most popular questions from this chapter

Solve each problem. Emission of Pollutants When a thermal inversion layer is over a city (as happens in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 A.M. Assume that the pollutant disperses horizontally over a circular area. If \(t\) represents the time, in hours, since the factory began emitting pollutants \((t=0 \text { represents } 8 \text { A.M.), assume that the radius of the circle of pollutants at time } t\) is \(r(t)=2 t\) miles. Let \(\mathscr{A}(r)=\pi r^{2}\) represent the area of a circle of radius \(r\) (a) Find \((\mathscr{A} \circ r)(t)\) (b) Interpret \((\mathscr{A} \circ r)(t)\) (c) What is the area of the circular region covered by the layer at noon?

In Exercises 27 and \(28,\) use the table to evaluate each expression in parts ( \(a\) )- \((d),\) if possible. (a) \((f+g)(2)\) (b) \((f-g)(4)\) (c) \((f g)(-2)\) (d) \(\left(\frac{f}{g}\right)(0)\) $$\begin{array}{|c|c|c|}\hline x & f(x) & g(x) \\\\\hline-2 & -4 & 2 \\\\\hline 0 & 8 & -1 \\\\\hline 2 & 5 & 4 \\\\\hline 4 & 0 & 0 \\\\\hline\end{array}$$

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=\sqrt{x+2}, \quad g(x)=-\frac{1}{x}$$

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=\sqrt{x}, \quad g(x)=\frac{3}{x+6}$$

Solve each problem. The cost to hire a caterer for a party depends on the number of guests attending. If 100 people attend, the cost per person will be 20 dollars . For each person less than \(100,\) the cost will increase by $$ 5 .\( Assume that no more than 100 people will attend. Let \)x\( represent the number less than 100 who do not attend. For example, if 95 attend, \)x=5\( (a) Write a function defined by \)N(x)\( giving the number of guests. (b) Write a function defined by \)G(x)\( giving the cost per guest. (c) Write a function defined by \)N(x) \cdot G(x)\( for the total cost, \)C(x)$ (d) What is the total cost if 80 people attend?
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