Use the vertical line test to identify graphs in which y is a function of x.
Table of contents
- 0. Review of Algebra4h 18m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations1h 43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 5m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 22m
- 10. Combinatorics & Probability1h 45m
- Appendix 1. Review of Real Numbers2h 24m
- Appendix 2. Linear Equations and Inequalities3h 42m
- OLD 9. Sequences, Induction, and Probability Coming soon
- 1. - OLD - Fundamental Concepts of Algebra Coming soon
- 2. - OLD - Equations and Inequalities Coming soon
- OLD 4. Rational Functions Coming soon
- OLD 2. Functions & Graphs Coming soon
- OLD 6. Exponential and Logarithmic Functions Coming soon
- OLD 7. Systems of Equations and Inequalities Coming soon
- OLD 8. Matrices and Determinants Coming soon
- OLD 9. Conic Sections Coming soon
2. Graphs of Equations
Graphs and Coordinates
Problem 27b
Textbook Question
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.f(x)=4x+5 b. f(x + 1)
Verified step by step guidance1
Step 1: Start with the given function f(x) = 4x + 5. This is a linear function where 'x' is the independent variable.
Step 2: Substitute (x + 1) into the function in place of 'x'. This means replacing every occurrence of 'x' in the function with (x + 1). The new expression becomes f(x + 1) = 4(x + 1) + 5.
Step 3: Apply the distributive property to simplify 4(x + 1). Multiply 4 by both 'x' and '1', resulting in 4x + 4.
Step 4: Combine like terms. Add the constant terms 4 and 5 together to simplify the expression further. This gives f(x + 1) = 4x + 9.
Step 5: The simplified expression for f(x + 1) is now f(x + 1) = 4x + 9. This is the final simplified form of the function evaluated at (x + 1).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Evaluation
Function evaluation involves substituting a specific value for the independent variable in a function. In this case, to evaluate f(x + 1), you replace x in the function f(x) = 4x + 5 with (x + 1). This process allows you to find the output of the function for a new input.
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Algebraic Simplification
Algebraic simplification is the process of reducing an expression to its simplest form. After substituting x + 1 into the function, you will need to simplify the resulting expression, combining like terms and performing any necessary arithmetic operations to achieve a clearer representation of the function's output.
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Linear Functions
A linear function is a polynomial function of degree one, which can be expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. The function f(x) = 4x + 5 is linear, indicating that its graph is a straight line. Understanding the properties of linear functions is essential for evaluating and interpreting their behavior.
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