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Questions and Answers
If f(x) = 1/9 - 2, what is f^-1(x)?
f^-1(x) = 1/9x + 2
In which year was the actual population of Center City most different from the value predicted by the linear model created using the data from 1990 and 2005?
Answer not provided.
Which characteristic of a data set makes a linear regression model unreasonable?
A correlation coefficient close to 0
If f(x) = 7 + 4x and g(x) = 1/2x, what is the value of (f/g)(5)?
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What is the average speed of a distance runner during the interval 0.75 to 1.00 hours?
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Which graph represents the same relation as the table below? (-2,5), (0,1), (1,-1), (2,-3)
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Given f(x) = 17 - x^2, what is the average rate of change in f(x) over the interval [1, 5]?
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If f(x) and its inverse function, f^-1(x), are both plotted on the same coordinate plane, where is their point of intersection?
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If u(x) = x^5 - x^4 + x^2 and v(x) = -x^2, which expression is equivalent to (u/v)(x)?
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Which of the following is an odd function?
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Give f(x) = 10 - 2x, find f(7).
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Which graph represents an odd function?
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If h(x) is the inverse of f(x), what is the value of h(f(x))?
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If f(x) = x + 9 and g(x) = -6, what describes the value of (f + g)(x)?
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The cost, c, of a ham sandwich at a deli varies directly with the number of sandwiches, n. If c = $54 when n is 9, what is the cost of the sandwiches when n is 3?
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If a(x) and b(x) are linear functions with one variable, which of the following expressions produces a quadratic function?
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Which of the following is an even function?
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Shayla's method of testing whether a graph of a relation is also a function by using the y-axis is valid.
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About what distance in feet would a person be 8 seconds after the experiment begins?
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If f(x) is an even function and (6, 8) is a point on the graph of f(x), what reason explains why (-6, 8) must also be a point on the graph?
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Which function has an inverse that is also a function?
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The value, V(m), of a comic book m months after publication has an average rate of change of -0.04 between m = 36 and m = 60. What statement must be true?
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Which scatterplot has a correlation coefficient closest to r = -1?
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Given f(x) = 3x - 1 and g(x) = 2x - 3, for which value of x does g(x) = f(2)?
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For what interval is the value of (f - g)(x) negative?
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Study Notes
Function Inverses and Composition
- If f(x) = 1/9 - 2, then the inverse function f⁻¹(x) is f⁻¹(x) = 1/9x + 2.
- For any function f(x) and its inverse h(x), the composition h(f(x)) equals f(x).
Linear Models and Predictions
- Population modeling of Center City can use data points from 1990 and 2005 to create a linear equation for population prediction.
- The year with actual population significantly differing from the predicted value must be determined through comparative analysis.
Characteristics of Data Sets
- A linear regression model is unreasonable when the correlation coefficient is close to 0, indicating little to no linear relationship.
Function Calculations and Values
- Evaluating functions: For f(x) = 7 + 4x and g(x) = 1/2x, the value of (f/g)(5) is calculated to be 270.
- Average speed during an interval can be calculated using distance traveled; for 0.75 to 1.00 hours, it is 5 mph.
Graph Representation and Relationships
- Points plotted on a graph can be matched to tables; the points (-2,5), (0,1), (1,-1), and (2,-3) correlate with graph D.
- An even function property implies that for any point (a, b) on the graph, the point (-a, b) must also be on the graph.
Function Properties and Identifications
- An odd function reflects symmetry about the origin; f(x) = -x exemplifies this property.
- For g(x) = 2x² + 1, this function is classified as an even function since f(-x) = f(x).
Mathematical Relationships and Cost
- Costs can be modeled using direct variation; if c = $54 for n = 9, then for n = 3, c = $18.
- The expression (ab)(x), where a(x) and b(x) are linear functions, results in a quadratic function.
Evaluating Graphs as Functions
- A relation’s graph can be assessed for function status by checking the y-axis intersections; only one intersection indicates a function.
- Analysis of scatterplots helps determine correlation; scatterplot D reflects a correlation coefficient closest to r = -1.
Rate of Change and Function Intersection
- The value of a comic book decreases on average by $0.04 each month between the specified months.
- The interval when (f - g)(x) is negative can be identified as (-∞, 2), indicating where the function f lies below g.
Studying That Suits You
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Description
Test your understanding of key mathematical concepts with these flashcards designed for Exam 1 preparation. Topics include function inverses, population modeling, and characteristics of data sets. Perfect for revising essential skills in algebra and statistics.
