Podcast
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?
- It becomes equal.
- It is ignored.
- It reverses. (correct)
- It remains the same.
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?
- It passes the horizontal line test. (correct)
- A one-to-one function has multiple outputs for the same input.
- It can have repeated elements in the range.
- It can have values in the domain paired with multiple elements in the range.
What is the first step in finding the inverse of a function?
What is the first step in finding the inverse of a function?
- Replace f(x) with y. (correct)
- Solve for x.
- Swap x and y.
- Graph the 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?
Which property is true for logarithmic functions?
Which property is true for logarithmic functions?
What defines an exponential inequality's solution method?
What defines an exponential inequality's solution method?
What indicates that an inverse function does not exist?
What indicates that an inverse function does not exist?
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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