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Chapter 1: Problem 30

Evaluate the function for the indicated values. \(g(x)=-7[x+4]+6\) (a) \(g\left(\frac{1}{8}\right)\) (b) \(g(9)\) (c) \(g(-4)\) (d) \(g\left(\frac{3}{2}\right)\)

Short Answer

Expert verified
The evaluated values are: (a) \(-21\frac{1}{4}\), (b) \(-85\), (c) \(6\), (d) \(-26\frac{1}{2}\)

Step by step solution

01

Evaluate the Function for \(x = \frac{1}{8}\)

We substitute \(\frac{1}{8}\) into the function \(g(x) = -7[x + 4] + 6\). This gives us: \(-7\left[\frac{1}{8} + 4\right] + 6 = -7\left[\frac{33}{8}\right] + 6 = -28\frac{1}{8} + 6 = -21\frac{1}{4}\)
02

Evaluate the Function for \(x = 9\)

We substitute \(9\) into the function \(g(x)\). This gives us: \(-7[9 + 4] + 6 = -7[13] + 6 = -85\)
03

Evaluate the Function for \(x = -4\)

We substitute \(-4\) into the function \(g(x)\). This gives us: \(-7[-4 + 4] + 6 = -7[0] + 6 = 6\)
04

Evaluate the Function for \(x = \frac{3}{2}\)

We substitute \(\frac{3}{2}\) into the function \(g(x)\). This gives us: \(-7\left[\frac{3}{2} + 4\right] + 6 = -7\left[\frac{11}{2}\right] + 6 = -38\frac{1}{2} + 6 = -26\frac{1}{2}\)

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Key Concepts

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

Piecewise Functions
In mathematics, piecewise functions are functions that have different expressions or formulas based on the input value of the variable. Depending on the condition that the variable satisfies, the function evaluates differently. Let's consider a simple example to illustrate this:
  • For example, a function might be defined as \(f(x) = x^2\) if \(x \leq 2\) and \(f(x) = 2x + 1\) if \(x > 2\).
  • This means that if the input, \(x\), is less than or equal to 2, you will use \(f(x) = x^2\) to find the function's value.
  • If \(x\) is greater than 2, you switch to the other formula, \(f(x) = 2x + 1\).

Piecewise functions are useful in various scenarios, such as computer graphics, economics, and situations where a process changes at certain thresholds.
Understanding how to determine which piece of the function to use is crucial when dealing with such functions. This makes them predictable yet adaptable for different situations.
Function Notation
Function notation is a way of representing mathematical functions in a precise and convenient form. It typically uses a letter, like \(f\), followed by the variable in parentheses, like \(f(x)\). This notation tells you:
  • The name of the function is \(f\).
  • \(x\) is the variable or input value of the function.
  • The output or value of the function is determined by plugging \(x\) into the expression named after \(f(x)\).

Function notation is hugely beneficial when evaluating functions for specific values. For example, if given \(f(x) = 3x + 5\) and you need \(f(2)\), you substitute 2 into the function's expression, leading to \(f(2) = 3(2) + 5 = 11\).
This notation clear specifies what operation is being performed on the variable. It helps you quickly identify inputs and the corresponding outputs within problems or real-world applications.
Substitution in Algebra
Substitution is a core concept in algebra that involves replacing a variable with a specific value to evaluate an expression or solve an equation. This technique is widely used in function evaluation, as shown in our initial problem.
  • First, you identify which value will replace the variable (such as \(x\)).
  • Next, you substitute this value directly into the expression.
  • Finally, you perform the arithmetic calculations to arrive at the result.

For instance, with the function \(g(x) = -7[x + 4] + 6\), to evaluate \(g\left(\frac{1}{8}\right)\), substitute \(\frac{1}{8}\) in place of \(x\). This gives \(-7\left[\frac{1}{8} + 4\right] + 6\).
This substitution process is useful for checking solutions and analyzing expressions by keeping your calculations organized and concise.

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Most popular questions from this chapter

The number \(N\) of bacteria in a refrigerated food is given by $$N(T)=10 T^{2}-20 T+600, \quad 2 \leq T \leq 20$$ where \(T\) is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by $$T(t)=3 t+2, \quad 0 \leq t \leq 6$$ where \(t\) is the time in hours. (a) Find the composition \((N \circ T)(t)\) and interpret its meaning in context. (b) Find the bacteria count after 0.5 hour. (c) Find the time when the bacteria count reaches 1500 .

The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width \(x\) surrounds the garden. (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter \(y\) of the walkway in terms of \(x\) (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one- meter increase in the width of the walkway, determine the increase in its perimeter.

For each function, sketch (on the same set of coordinate axes) a graph of each function for \(c=-1,1,\) and 3. (a) \(f(x)=|x|+c\) (b) \(f(x)=|x-c|\)

Write an equation for the function described by the given characteristics. The shape of \(f(x)=x^{3},\) but shifted 13 units to the right

You are driving on a road that has a \(6 \%\) uphill grade. This means that the slope of the road is \(\frac{6}{100} .\) Approximate the amount of vertical change in your position when you drive 200 feet.
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