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Chapter 5: Problem 12

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

Short Answer

Expert verified
Result: 42

Step by step solution

01

Evaluate f(-2)

To find the value of function \(f\) at \(x = -2\), substitute \(-2\) into the function \(f(x)\): \[f(-2) = -3(-2) + 1 = 6 + 1 = 7\].
02

Evaluate g(-2)

To find the value of function \(g\) at \(x = -2\), substitute \(-2\) into the function \(g(x)\): \[g(-2) = (-2)^2 + 2 = 4 + 2 = 6\].
03

Multiply the results

Multiply the values obtained from \(f(-2)\) and \(g(-2)\): \[f(-2) \times g(-2) = 7 \times 6 = 42\].

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Key Concepts

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

Linear Functions
A linear function is a function that creates a straight line when graphed on a coordinate plane. This means it has a constant rate of change. The general form of a linear function is given by: \(f(x) = mx + b\), where: - \(m\) is the slope, indicating the steepness of the line - \(b\) is the y-intercept, where the line crosses the y-axis For the given function, \(f(x) = -3x + 1\), we see that the slope is \(-3\) and the y-intercept is \(1\). In our solution, we evaluated \(f(-2)\) by plugging \(-2\) into \(f(x)\). This gives: \(f(-2) = -3(-2) + 1 = 6 + 1 = 7\). So, at \(x = -2\), the value of the linear function \(f(x)\) is \(7\).
Quadratic Functions
Quadratic functions represent parabolas on a graph. They have the general form: \(g(x) = ax^2 + bx + c\), where: - \(a\), \(b\), and \(c\) are constants The coefficient \(a\) determines the direction of the parabola (opens upwards if \(a > 0\) and downwards if \(a < 0\)). In our function \(g(x) = x^2 + 2\), \(a = 1\), \(b = 0\), and \(c = 2\), giving us an upward-opening parabola. We calculated \(g(-2)\) by substituting \(-2\) into \(g(x)\): \(g(-2) = (-2)^2 + 2 = 4 + 2 = 6\). This tells us that the value of \(g(x)\) at \(x = -2\) is \(6\).
Substitution Method
The substitution method is a technique used to evaluate functions. You replace the variable with a given number and then solve. It's straightforward but requires careful calculation. Here鈥檚 a step-by-step breakdown: - Identify the value you need to substitute - Replace \(x\) in the function with this value - Simplify the expression to find the result In our exercise, we used this method to find \(f(-2)\) and \(g(-2)\). By substituting \(-2\) into \(f(x)\): \(f(-2) = -3(-2) + 1 = 6 + 1 = 7\). And substituting \(-2\) into \(g(x)\): \(g(-2) = (-2)^2 + 2 = 4 + 2 = 6\).
Multiplication of Functions
Multiplication of functions involves finding the product of the values obtained from two functions. If you compute \(f(a)\) and \(g(a)\), the multiplication of these functions at \(a\) is performed as follows: \([f \times g](a) = f(a) \times g(a)\). This is useful when combining results from different functional evaluations. For our specific problem, once we determined: - \(f(-2) = 7\) - \(g(-2) = 6\) We multiplied these values to find: \(f(-2) \times g(-2) = 7 \times 6 = 42\). Hence, the final result of multiplying these function values is \(42\).

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Most popular questions from this chapter

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{1}{x-3}\\\ &g(x)=4 x^{3} \end{aligned} $$

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{7 x}{x-2}\\\ &g(x)=3 x+7 \end{aligned} $$

Find the domain of \(F / G,\) if $$ F(x)=\frac{1}{x-4} \quad \text { and } \quad G(x)=\frac{x^{2}-4}{x-3} $$

Write \(\frac{4}{32}\) in decimal notation, simplified fraction notation, and scientific notation.

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{2} \square 4^{3} $$
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