Podcast
Questions and Answers
Given $f(x) = -x^2 + 3x + 5$ and $g(x) = \frac{-2x}{1-x}$, what is the value of $f(g(-3))$?
Given $f(x) = -x^2 + 3x + 5$ and $g(x) = \frac{-2x}{1-x}$, what is the value of $f(g(-3))$?
- -45.25 (correct)
- -55
- -67.75
- Does not exist
If $f(x) = -x^2 + 3x + 5$ and $g(x) = \frac{-2x}{1-x}$, what is the expression for $(g \circ f)(x)$?
If $f(x) = -x^2 + 3x + 5$ and $g(x) = \frac{-2x}{1-x}$, what is the expression for $(g \circ f)(x)$?
- $\frac{-2x^2 + 6x + 10}{x^2 - 3x - 4}$
- $\frac{2x^2 - 6x - 10}{x^2 - 3x - 4}$ (correct)
- $\frac{2x^2 + 6x + 10}{x^2 - 3x - 6}$
- $\frac{-2x^2 + 6x + 10}{x^2 + 3x + 4}$
Given $f(x) = -x^2 + 3x + 5$ and $g(x) = \frac{-2x}{1-x}$, what is the domain of $(g \circ f)(x)$?
Given $f(x) = -x^2 + 3x + 5$ and $g(x) = \frac{-2x}{1-x}$, what is the domain of $(g \circ f)(x)$?
- $\mathbb{R}, x \neq 1, 4$
- $\mathbb{R}, x \neq -1, -4$
- $\mathbb{R}, x \neq -4, 1$
- $\mathbb{R}, x \neq -1, 4$ (correct)
What is the inverse function of $f(x) = 7x - 8$?
What is the inverse function of $f(x) = 7x - 8$?
Given the graph of $f(x)$ and $g(x)$, what is $(g \circ f)(-1)$?
Given the graph of $f(x)$ and $g(x)$, what is $(g \circ f)(-1)$?
Given the graph of $f(x)$, what is $f^{-1}(3)$?
Given the graph of $f(x)$, what is $f^{-1}(3)$?
If $f(x) = \frac{2}{x-1}$ and $g(x) = \frac{x+2}{x}$, which statement correctly justifies whether they are inverses of each other?
If $f(x) = \frac{2}{x-1}$ and $g(x) = \frac{x+2}{x}$, which statement correctly justifies whether they are inverses of each other?
Expand the following expression using properties of logarithms: $\log(\frac{x^2}{yz^3})$
Expand the following expression using properties of logarithms: $\log(\frac{x^2}{yz^3})$
Using properties of logarithms, write the following expression as a single logarithm: $3\ln(x) - 2\ln(z) + \ln(y)$
Using properties of logarithms, write the following expression as a single logarithm: $3\ln(x) - 2\ln(z) + \ln(y)$
For the equation $20e^{2x}=300$, which of the following represents the exact value of $x$?
For the equation $20e^{2x}=300$, which of the following represents the exact value of $x$?
Flashcards
What is composite function?
What is composite function?
A function resulting from applying one function to the results of another.
What is the domain of a function?
What is the domain of a function?
The set of all possible input values (x-values) for which a function is defined.
What is an inverse function?
What is an inverse function?
A function that 'undoes' the action of another function.
What is a logarithmic equation?
What is a logarithmic equation?
Signup and view all the flashcards
What is the y-intercept?
What is the y-intercept?
Signup and view all the flashcards
Study Notes
Composite Functions and Evaluation
- Given f(x) = -x² + 3x + 5 and g(x) = -2/(1-x), evaluate f(g(-3)) by first finding g(-3) and then substituting that value into f(x).
- To find the expression for (g ◦ f)(x), substitute f(x) into g(x).
- Determine the domain of (g ◦ f)(x) by considering the restrictions on the domain of f(x) and the domain of the composite function.
Inverse Functions
- To find the inverse function of f(x) = 7x - 8, swap x and y and solve for y.
- To determine if two functions f(x) = 2/(x-1) and g(x) = (x+2)/x are inverses, verify if f(g(x)) = x and g(f(x)) = x.
Graph Interpretation
- Evaluate (g ◦ f)(-1) using the provided graph by finding f(-1) from the graph then using the result as input for g(x).
- Determine f⁻¹(3) from the graph by finding the x-value for which f(x) = 3.
Logarithmic Properties: Expansion
- Expand log(x²/(yz³)) using logarithm properties, resulting in 2log(x) - log(y) - 3log(z).
Logarithmic Properties: Condensation
- Combine 3ln(x) - 2ln(z) + ln(y) into a single logarithm: ln((x³y)/z²).
Logarithmic Functions: Analysis
- For a logarithmic function g(x) = log(2 - 3x), find the y-intercept by setting x=0 and solving for g(0).
- The domain of g(x) = log(2 - 3x) is found by setting 2 - 3x > 0 and solving for x.
- Find the inverse function g⁻¹(x) by swapping x and y in the equation y = log(2 - 3x) and solving for y.
- Determine the y-intercept of g⁻¹(x) by setting x = 0 in the inverse function and solving for y.
- Determine the domain of g⁻¹(x) by analyzing any restrictions on the inverse function.
Exponential Equations: Solving
- To solve an exponential equation like 20e^(2x) = 300, isolate the exponential term and take the natural logarithm of both sides.
- To solve 7log₅(1 - 3x) = 25, isolate the logarithm and rewrite the equation in exponential form.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
