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Chapter 3: Problem 55

If \(g(x)=10 x\), evaluate \(g(40)\).

Short Answer

Expert verified
g(40) = 400.

Step by step solution

01

Identify the given function

The function given is \(g(x) = 10x\). This is a linear function where every input \(x\) is multiplied by 10.
02

Substitute the value into the function

To find \(g(40)\), substitute 40 for \(x\) in the function. This means replacing \(x\) with 40 in \(g(x) = 10x\).
03

Perform the multiplication

Now calculate \(g(40)\) by performing the multiplication: \(10 \times 40\).
04

Write the final answer

The result of the multiplication is 400. Thus, \(g(40) = 400\).

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Key Concepts

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

linear functions
Linear functions are a type of mathematical function where each input value has a corresponding output value that forms a straight line when graphed on a coordinate plane.
In simpler terms, linear functions show a constant rate of change between the input and output values.
If you have a function written as:
\(f(x) = mx + b\)
Here, \(m\) is the slope (or rate of change) and \(b\) is the y-intercept (the point where the line crosses the y-axis).
In our example, the function is \(g(x) = 10x\), which means every input \(x\) is multiplied by 10.
There is no y-intercept \(b\) because it is zero.
function notation
Function notation is a way to write functions in mathematics.
It uses symbols to represent the relationship between inputs and outputs.
The most common notation is \(f(x)\), where \(f\) is the function and \(x\) is the input variable.
For example, if we have \(f(x) = 3x + 2\), then \(f(2)\) means we substitute \(x\) with 2, making it \(f(2) = 3(2) + 2\).
In our given function, \(g(x) = 10x\), substituting 40 for \(x\) changes it to \(g(40) = 10(40)\).
This step is crucial because it tells us how to evaluate the function at specific points.
substitution
Substitution is the process of replacing a variable in an expression with a specific value.
In the context of functions, substitution helps us find the output for a given input by replacing the variable with the given number.
For example, in the function \(g(x) = 10x\), if we want to find \(g(40)\), we substitute 40 for \(x\).
This process is straightforward:
  • Identify the given function and the input value you need to substitute.
  • Replace the variable \(x\) with that input value.
  • Perform any necessary calculations to find the output.

Let's go through the solution:
  • Given function: \(g(x) = 10x\).
  • Requested input: 40.
  • Perform substitution: \(g(40) = 10(40)\).
  • Perform multiplication: \(10 \times 40 = 400\).
Hence, the final answer is \(g(40) = 400\).
This simple method helps us easily evaluate functions.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \(\left(-8, \frac{1}{4}\right) ;\left(16, \frac{1}{2}\right)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(-5 x+2 y=40\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((0,-5) ;(2,0)\)

For exercises 103-104, some learning preferences describe how you prefer to receive, think about, and learn new information. These preferences include visual learning, auditory learning, and kinesthetic learning. Many students use more than one of these categories as they learn mathematics. \- Visual learners prefer to see information. Although you definitely listen to your instructor, you also like to see the example on a white board or screen. You may be able to recall a process by visualizing it in your mind; you may learn better by organizing information in charts, tables, diagrams, or pictures. You may prefer the use of colored markers instead of just black. \- Auditory learners prefer to hear information. Although you definitely watch what your instructor is doing, you also like your instructor to explain things aloud as he or she works. You may find it difficult to take notes because you cannot concentrate enough on what is being said while you write. You may learn better if you have the chance to explain things to others. \- Kinesthetic learners prefer to do. You may find it difficult to sit still and just watch and listen; you want to be trying it out. You may find that you must take notes in order to learn. If you only watch and listen, you may understand the concept but not remember it after you leave the classroom. You often learn better if you can show others how to do things. Do you have a strong preference for visual, auditory, or kinesthetic learning?

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-1,-3) ;(-4,-1)\)
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