Math Exam 1 Flashcards
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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)?

<p>270</p> Signup and view all the answers

What is the average speed of a distance runner during the interval 0.75 to 1.00 hours?

<p>5 mph</p> Signup and view all the answers

Which graph represents the same relation as the table below? (-2,5), (0,1), (1,-1), (2,-3)

<p>Graph D (C)</p> Signup and view all the answers

Given f(x) = 17 - x^2, what is the average rate of change in f(x) over the interval [1, 5]?

<p>-6</p> Signup and view all the answers

If f(x) and its inverse function, f^-1(x), are both plotted on the same coordinate plane, where is their point of intersection?

<p>(2, 2)</p> Signup and view all the answers

If u(x) = x^5 - x^4 + x^2 and v(x) = -x^2, which expression is equivalent to (u/v)(x)?

<p>-x^3 + x^2 - 1</p> Signup and view all the answers

Which of the following is an odd function?

<p>f(x) = -x (C)</p> Signup and view all the answers

Give f(x) = 10 - 2x, find f(7).

<p>-4</p> Signup and view all the answers

Which graph represents an odd function?

<p>Graph A (A)</p> Signup and view all the answers

If h(x) is the inverse of f(x), what is the value of h(f(x))?

<p>f(x)</p> Signup and view all the answers

If f(x) = x + 9 and g(x) = -6, what describes the value of (f + g)(x)?

<p>(f + g)(x) &gt;= 3 for all values of x</p> Signup and view all the answers

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?

<p>$18</p> Signup and view all the answers

If a(x) and b(x) are linear functions with one variable, which of the following expressions produces a quadratic function?

<p>(ab)(x)</p> Signup and view all the answers

Which of the following is an even function?

<p>g(x) = 2x^2 + 1 (D)</p> Signup and view all the answers

Shayla's method of testing whether a graph of a relation is also a function by using the y-axis is valid.

<p>False (B)</p> Signup and view all the answers

About what distance in feet would a person be 8 seconds after the experiment begins?

<p>27 ft</p> Signup and view all the answers

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?

<p>Since the function is even, the outputs of a negative x-value and a positive x-value are the same.</p> Signup and view all the answers

Which function has an inverse that is also a function?

<p>g(x) = 2x - 3</p> Signup and view all the answers

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?

<p>The value of the comic book decreased by an average of $0.04 each month between m = 36 and m = 60.</p> Signup and view all the answers

Which scatterplot has a correlation coefficient closest to r = -1?

<p>Scatterplot D (C)</p> Signup and view all the answers

Given f(x) = 3x - 1 and g(x) = 2x - 3, for which value of x does g(x) = f(2)?

<p>x = 4</p> Signup and view all the answers

For what interval is the value of (f - g)(x) negative?

<p>(-∞, 2)</p> Signup and view all the answers

Flashcards

Inverse Function

A function's inverse reverses the input and output. If f(x) takes x to y, then f⁻¹(x) takes y back to x.

Composition of Inverse Functions

The composition of a function and its inverse always results in the original input.

Linear Model for Population

A linear model is a straight line used to represent data. It's created using two data points and its equation represents the relationship between variables.

Correlation Coefficient and Linear Regression

The correlation coefficient measures the strength and direction of a linear relationship. A value near 0 indicates a weak or no linear relationship.

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Evaluating Functions

To evaluate a function, substitute the given value of x into the function's formula and simplify.

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Average Speed Calculation

Average speed is calculated by dividing the total distance traveled by the time taken.

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Matching Graph Points with Table Values

Points plotted on a graph can be matched to their corresponding values in a table.

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Even Function Property

An even function is symmetric about the y-axis. If (a, b) is on the graph, then (-a, b) is also on the graph.

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Odd Function Property

An odd function is symmetric about the origin. If (a, b) is on the graph, then (-a, -b) is also on the graph.

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Direct Variation

Direct variation describes a relationship where one variable is proportional to another. If y varies directly with x, then y = kx, where k is the constant of variation.

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Product of Linear Functions

The product of two linear functions (a(x) and b(x)) results in a quadratic function.

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Vertical Line Test

To determine if a graph represents a function, check if every vertical line intersects the graph at most once.

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Scatterplot and Correlation

Scatterplots show the relationship between two variables. A strong negative correlation means the points form a downward trend.

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Rate of Change

The average rate of change between two points is the change in y divided by the change in x. It represents the slope of the line connecting the points.

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Function Subtraction and Inequality

To find when (f - g)(x) is negative, identify the intervals where the graph of f(x) lies below the graph of g(x).

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Finding Inverse Function

To determine the inverse function, switch the roles of x and y in the original function and solve for y.

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Composition of Inverse Functions Property

For any function f(x) and its inverse h(x), the composition f(h(x)) equals x.

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Function Addition and Intersection

The value of x when (f + g)(x) = 0 is the intersection point between the graphs of f(x) and g(x).

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Equation of a Line

To find the equation of the line that passes through two points, calculate the slope and use the point-slope form of the equation: y - y₁ = m(x - x₁).

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Linear Model Validity

A linear model is appropriate when the data shows a clear linear relationship, with points clustering closely around a line.

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Linear Model Prediction

To predict values using a linear model, substitute the desired x-value into the linear equation to find the corresponding y-value.

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Y-intercept in Linear Model

The y-intercept of a linear model represents the initial value or starting point for the relationship being modeled.

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Slope in Linear Model

The slope of a linear model represents the rate of change of the dependent variable with respect to the independent variable.

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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.

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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.

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