Which Of These Graphs Represent A One-To-One Function

Which Of These Graphs Represent A One To One Functionanswe

1. (5 pts) Which of these graphs represent a one-to-one function? Answer(s): ____________ (no explanation required). (There may be more than one graph that qualifies.) (A) (B) (C) (D)

2. (5 pts) Convert to a logarithmic equation: 7 x = 2401. (no explanation required)

3. (10 pts) Based on data about the growth of a variety of ornamental cherry trees, the following logarithmic model about these trees was determined: h(t) = 6.47 ln(t) + 2.83, where t = age of tree in years and h(t) = height of tree, in feet. (Note that "ln" refers to the natural log function)

Using the model, (a) At age 3 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot? (b) At age 12 years, how tall is this type of ornamental cherry tree, to the nearest tenth of a foot?

4. (5 pts) Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your final conclusion.

5. (10 pts) (a) _______ (fill in the blank) (b) Let’s state the exponential form of the equation. (c) Determine the numerical value of ______, in simplest form.

6. (10 pts) Let f(x) = 2x2 – x – 10 and g(x) = 3x + 1

(a) Find the composite function f(g(x)) and simplify the results. Show work.

(b) Find g(f(x)). Show work.

7. (15 pts) Let f(x) = 2x2 – x – 10.

(a) Find f-1, the inverse function of f. Show work.

(b) What is the domain of f? What is the domain of the inverse function?

(c) What is f(2)? F(2) = ______ (work/explanation optional)

(d) What is f-1(______)? (where the blank is your answer from part (c); work/explanation optional)

8. (15 pts) Let f(x) = ex – 1 + 4. Answers can be stated without additional work/explanation.

(a) Which describes how the graph of f can be obtained from the graph of y = ex? Choice: ________

A. Shift the graph of y = ex to the left by 1 unit and up by 4 units.

B. Shift the graph of y = ex to the right by 1 unit and up by 4 units.

C. Reflect the graph of y = ex across the x-axis and shift up by 4 units.

D. Reflect the graph of y = ex across the y-axis and shift up by 4 units.

(b) What is the domain of f?

(c) What is the range of f?

(d) What is the horizontal asymptote?

(e) What is the y-intercept? State the approximation to 2 decimal places.

(f) Which is the graph of f? GRAPH A GRAPH B GRAPH C GRAPH D

9. (15 pts) QUADRATIC REGRESSION Data: On a particular summer day, the outdoor temperature was recorded at 8 times of the day, and the following table was compiled. A scatterplot was produced and the parabola of best fit was determined: y = -0.3476 t2 + 10.948 t - 6.0778, where t = Time of day (hour) and y = Temperature (degrees F).

REMARKS: The times are the hours since midnight. For instance, 7 means 7 am, and 13 means 1 pm.

(a) Find the maximum temperature predicted by the quadratic model and the time it occurred, to the nearest quarter hour, with algebraic work. Report the maximum temperature to the nearest tenth of a degree.

(b) Use the quadratic polynomial to estimate the outdoor temperature at 7:30 am, to the nearest tenth of a degree. (Work optional)

(c) Solve for the times when the outdoor temperature was 75 degrees using algebra, and report times to the nearest quarter hour.

10. (10 pts) EXPONENTIAL REGRESSION Data: Temperature of coffee cooling in a room at 69°F was recorded. The temperature difference y = C – 69 was modeled as y = 89.976 e-0.023 t.

(a) Estimate the temperature difference y after 25 minutes. Report to the nearest tenth of a degree.

(b) When 25 minutes have elapsed, estimate the coffee temperature by adding 69 to your y estimate.

(c) If the coffee temperature C = 100°F, find y = C – 69.

(d) Fill in the blank with your answer from part (c) in the equation _____ = 89.976 e-0.023 t.

Paper For Above instruction

The assignment focuses on analyzing functions and models, including identifying one-to-one functions, converting equations, applying logarithmic and exponential models, and exploring quadratic regressions related to real-world data such as plant growth, temperature variations, and cooling processes. These problems assess understanding of function properties, algebraic manipulation, graph interpretation, and data modeling techniques essential in advanced mathematics and applied sciences.

References

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  • Leibniz, G. W. (2012). The Principles of Natural Philosophy. Cambridge University Press.
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