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Questions and Answers
What happens to the inequality sign when solving an inequality with a base between 0 and 1?
What happens to the inequality sign when solving an inequality with a base between 0 and 1?
Which of the following statements is true about a one-to-one function?
Which of the following statements is true about a one-to-one function?
What is the first step in finding the inverse of a function?
What is the first step in finding the inverse of a function?
How do you solve exponential equations that have bases that are the same?
How do you solve exponential equations that have bases that are the same?
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Which property is true for logarithmic functions?
Which property is true for logarithmic functions?
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What defines an exponential inequality's solution method?
What defines an exponential inequality's solution method?
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What indicates that an inverse function does not exist?
What indicates that an inverse function does not exist?
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When analyzing a mapping diagram, which scenario indicates a one-to-one function?
When analyzing a mapping diagram, which scenario indicates a one-to-one function?
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Study Notes
Solving Inequalities with Exponential Bases
- When solving inequalities with exponential bases, the inequality sign remains the same if the base is greater than 1.
- If the base is between 0 and 1, the inequality sign reverses.
- The section provides step-by-step solutions for various examples.
One-to-One Functions
- A one-to-one function maps each element in the domain to a unique element in the range, and vice versa.
- Mapping diagrams visually represent one-to-one and non-one-to-one functions.
- The horizontal line test determines if a function is one-to-one graphically.
Inverse Functions
- Inverse functions, denoted by f⁻¹(x), reverse the input and output of a function.
- Finding the inverse function involves:
- Replacing f(x) with y.
- Swapping x and y.
- Solving for y.
- Replacing y with f⁻¹(x).
- If the original function fails the horizontal line test, the inverse function does not exist.
Exponential Equations
- Exponential equations involve variables in exponents.
- The one-to-one property of exponential functions states that if the bases are the same, the exponents must be equal.
- Solving exponential equations involves making the bases the same and equating the exponents.
Logarithmic Functions
- Logarithmic functions are the inverse of exponential functions.
- They have specific domain, range, vertical asymptote, and intercepts.
- Plotting a logarithmic function allows for determining its domain, range, intercepts, and zeros.
Exponential Inequalities
- Exponential inequalities involve comparing exponential expressions.
- The rules for solving depend on whether the base of the exponential expression is greater than 1 or between 0 and 1, similar to solving inequalities with exponential bases.
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Description
This quiz covers key concepts of solving inequalities with exponential bases, identifying one-to-one functions, and finding inverse functions. You'll explore graphical representations and mathematical procedures to enhance your understanding of these essential algebra topics.
