For The Following Exercise, Evaluate The Function { F $}$ At The Indicated Values: { F(3), F(-2), F(-a), -f(a), F(a+h) $}$. Simplify Your Results.Given: ${ f(x) = 2x - 5 } E V A L U A T E A N D S I M P L I F Y : Evaluate And Simplify: E V A L U A T E An D S Im Pl I F Y : [ f(3) = \square

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In this exercise, we are given a function f(x) = 2x - 5 and asked to evaluate it at various values. We will simplify our results and discuss the process of evaluating functions at specific values.

Evaluating f(3)

To evaluate f(3), we substitute x = 3 into the function f(x) = 2x - 5.

f(3) = 2(3) - 5 = 6 - 5 = 1

Therefore, f(3) = 1.

Evaluating f(-2)

To evaluate f(-2), we substitute x = -2 into the function f(x) = 2x - 5.

f(-2) = 2(-2) - 5 = -4 - 5 = -9

Therefore, f(-2) = -9.

Evaluating f(-a)

To evaluate f(-a), we substitute x = -a into the function f(x) = 2x - 5.

f(-a) = 2(-a) - 5 = -2a - 5

Therefore, f(-a) = -2a - 5.

Evaluating -f(a)

To evaluate -f(a), we multiply the function f(x) = 2x - 5 by -1.

-f(a) = -(2a - 5) = -2a + 5

Therefore, -f(a) = -2a + 5.

Evaluating f(a + h)

To evaluate f(a + h), we substitute x = a + h into the function f(x) = 2x - 5.

f(a + h) = 2(a + h) - 5 = 2a + 2h - 5

Therefore, f(a + h) = 2a + 2h - 5.

Discussion

In this exercise, we evaluated the function f(x) = 2x - 5 at various values. We simplified our results and discussed the process of evaluating functions at specific values. The function f(x) = 2x - 5 is a linear function, and its graph is a straight line with a slope of 2 and a y-intercept of -5.

Conclusion

In conclusion, evaluating functions at specific values is an important concept in mathematics. It allows us to find the output of a function for a given input and to understand the behavior of the function. In this exercise, we evaluated the function f(x) = 2x - 5 at various values and simplified our results. We hope that this exercise has helped you to understand the concept of evaluating functions at specific values.

Key Takeaways

  • Evaluating functions at specific values is an important concept in mathematics.
  • The function f(x) = 2x - 5 is a linear function with a slope of 2 and a y-intercept of -5.
  • To evaluate a function at a specific value, we substitute the value into the function and simplify the result.
  • The process of evaluating functions at specific values is used to find the output of a function for a given input and to understand the behavior of the function.

Real-World Applications

Evaluating functions at specific values has many real-world applications. For example, in physics, we use functions to model the motion of objects. In economics, we use functions to model the behavior of markets. In computer science, we use functions to model the behavior of algorithms.

Future Directions

In the future, we will explore more advanced concepts in mathematics, such as calculus and differential equations. We will also explore the applications of mathematics in various fields, such as physics, economics, and computer science.

References

  • [1] "Calculus" by Michael Spivak
  • [2] "Differential Equations" by Lawrence Perko
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

In this article, we will answer some common questions about evaluating functions. We will cover topics such as evaluating functions at specific values, simplifying expressions, and understanding the behavior of functions.

Q: What is evaluating a function?

A: Evaluating a function means finding the output of the function for a given input. In other words, it means substituting a value into the function and simplifying the result.

Q: How do I evaluate a function at a specific value?

A: To evaluate a function at a specific value, you need to substitute the value into the function and simplify the result. For example, if we have the function f(x) = 2x - 5 and we want to evaluate it at x = 3, we would substitute x = 3 into the function and simplify the result.

f(3) = 2(3) - 5 = 6 - 5 = 1

Q: What is the difference between evaluating a function and simplifying an expression?

A: Evaluating a function and simplifying an expression are two related but distinct concepts. Evaluating a function means finding the output of the function for a given input, while simplifying an expression means reducing a complex expression to its simplest form.

For example, if we have the expression 2x - 5, we can simplify it by combining like terms:

2x - 5 = 2x - 5

However, if we want to evaluate the function f(x) = 2x - 5 at x = 3, we would substitute x = 3 into the function and simplify the result:

f(3) = 2(3) - 5 = 6 - 5 = 1

Q: How do I know if a function is linear or non-linear?

A: A function is linear if it can be written in the form f(x) = ax + b, where a and b are constants. If a function cannot be written in this form, it is non-linear.

For example, the function f(x) = 2x - 5 is linear because it can be written in the form f(x) = ax + b, where a = 2 and b = -5.

Q: What is the significance of the slope of a linear function?

A: The slope of a linear function is a measure of how steep the function is. It is calculated by dividing the change in the output by the change in the input.

For example, if we have the function f(x) = 2x - 5, the slope is 2. This means that for every unit increase in the input, the output increases by 2 units.

Q: How do I graph a linear function?

A: To graph a linear function, you need to plot two points on the graph and draw a line through them. The slope of the line is the same as the slope of the function.

For example, if we have the function f(x) = 2x - 5, we can plot two points on the graph, such as (0, -5) and (1, -3), and draw a line through them. The slope of the line is 2, which is the same as the slope of the function.

Q: What are some common applications of linear functions?

A: Linear functions have many applications in real-world problems, such as:

  • Modeling the motion of objects
  • Describing the behavior of markets
  • Calculating the cost of goods
  • Determining the amount of time it takes to complete a task

Q: What are some common mistakes to avoid when evaluating functions?

A: Some common mistakes to avoid when evaluating functions include:

  • Not substituting the correct value into the function
  • Not simplifying the result correctly
  • Not checking for domain restrictions
  • Not considering the context of the problem

Conclusion

Evaluating functions is an important concept in mathematics that has many real-world applications. By understanding how to evaluate functions, you can solve problems in a variety of fields, from physics and economics to computer science and engineering. Remember to always substitute the correct value into the function, simplify the result correctly, and check for domain restrictions.

Key Takeaways

  • Evaluating a function means finding the output of the function for a given input.
  • Simplifying an expression means reducing a complex expression to its simplest form.
  • A function is linear if it can be written in the form f(x) = ax + b, where a and b are constants.
  • The slope of a linear function is a measure of how steep the function is.
  • Linear functions have many applications in real-world problems.

Real-World Applications

Evaluating functions has many real-world applications, including:

  • Modeling the motion of objects
  • Describing the behavior of markets
  • Calculating the cost of goods
  • Determining the amount of time it takes to complete a task

Future Directions

In the future, we will explore more advanced concepts in mathematics, such as calculus and differential equations. We will also explore the applications of mathematics in various fields, such as physics, economics, and computer science.

References

  • [1] "Calculus" by Michael Spivak
  • [2] "Differential Equations" by Lawrence Perko
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for informational purposes only and are not required for this exercise.