Math 107 Quiz 5 Name Instructor K Chavi

Math 107 Quiz 5name Instructor K Chavi

Convert to a logarithmic equation: 6 x = 7776.

(a) _______ (fill in the blank)

(b) Let State the exponential form of the equation.

(c) Determine the numerical value of , in simplest form. Work optional.

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)

A human memory model is used to determine the percentage, M(t), of information that students remember months after the completion of a course. For a specific geography course, students followed the model:

a) What percentage of material did the students remember after 8 months? Show work.

b) How many months after the semester did students still remember 45 percent of the material from the course? Show work.

Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your final conclusion.

Let f(x) = 3 x² – 4 x – 2 and g(x) = 2 x + 1:

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

(b) Find . Show work.

Let (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 number in the blank is your answer from part (c)? work/explanation optional

Let f(x) = e^x – 2 + 3. Answers can be stated without additional work/explanation.

(a) Which describes how the graph of f can be obtained from the graph of y = e^x? Choice: ________ A. Shrink the graph of y = e^x horizontally by a factor of 2 and shift up by 3 units. B. Reflect the graph of y = e^x across the x-axis and shift up by 1 unit. C. Shift the graph of y = e^x to the left by 2 units and up by 3 units. D. Shift the graph of y = e^x to the right by 2 units and up by 3 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 (i.e., the nearest hundredth).

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

In Exercises 1 - 33, solve the equation analytically.

Paper For Above instruction

The following paper addresses the comprehensive mathematical problems presented in the quiz, including conversions, graph interpretation, exponential decay modeling, solving equations, composite and inverse functions, transformations of exponential functions, and analysis of exponential functions’ properties.

1. Conversion of a exponential equation to a logarithmic form: given 6^x = 7776, the goal is to express x in terms of logarithms. Using the definition of logarithm, the equivalent logarithmic form is:

log₆ (7776) = x.

Expressed in natural logarithms, this becomes:

x = ln(7776) / ln(6).

Calculating this, ln(7776) ≈ 8.958, and ln(6) ≈ 1.792, therefore:

x ≈ 8.958 / 1.792 ≈ 5.

Thus, x is approximately 5, indicating that 6⁵ = 7776.

2. The problem involves a blank and the exponential form: dependent on the specific prompt, similar conversions apply, often involving writing the exponential form from a given logarithmic expression.

3. Identifying one-to-one graphs typically involves analyzing the graphs to see if they pass the horizontal line test, ensuring each horizontal line intersects the graph at most once. The graphs labeled A, B, C, D are analyzed; often, graphs with monotonically increasing or decreasing functions are one-to-one, while those with peaks or valleys are not.

4. The memory model involves an exponential decay function M(t), representing the percentage of information retained over time. For example, suppose M(t) = M₀ * e^-kt, where M₀ is initial memory percentage and k is a decay constant.

a) To find the percentage after 8 months:

substitute t=8 into the model, for example, if given M₀ = 100%, and decay constant k, then:

M(8) = 100 e^-k8. Assuming a decay rate, approximate the value numerically to find the remaining percentage.

b) To find the time when 45% of material remains:

set M(t) = 45 and solve for t:

45 = 100 * e^{-k t} => e^{-k t} = 0.45 => -k t = ln(0.45) => t = -ln(0.45)/k.

5. Solving the algebraic equation involves factoring or applying logarithms, depending on the form. For example, solving quadratic equations, applying quadratic formula or factoring methods will be used, followed by substitution to verify solutions.

6. Composite functions and evaluations:

Given f(x) = 3x² – 4x – 2 and g(x) = 2x + 1:

a) To find (f ◦ g)(x):

f(g(x)) = 3(2x+1)² – 4(2x+1) – 2.

The expansion yields: 3(4x² + 4x + 1) – 8x – 4 – 2 = 12x² + 12x + 3 – 8x – 6 = 12x² + 4x – 3.

b) To find the value of a specified composition or for other specific evaluations, substitute the value into the simplified expression accordingly.

7. Inverse functions:

a) To find f^–1(x), replace f(x) with y and solve for x in terms of y, then swap: y = 3x² – 4x – 2.

solving for x involves treating this as a quadratic in x and applying quadratic formula, leading to:

x = [4 ± √(16 – 12(y + 2))] / 6.

The inverse function is then expressed as a function of y, with the domain being the range of f, and vice versa.

b) Domain of f is all real numbers, since polynomial functions are defined everywhere. Domain of f^–1 is the range of f, which depends on the quadratic's vertex and y-values it attains.

c) To compute f(2):

f(2) = 3(4) – 8 – 2 = 12 – 8 – 2 = 2.

d) For the inverse value at f(2), plug in 2 into f^–1.

8. Analyzing exponential functions of the form f(x) = e^x – 2 + 3, which simplifies to f(x) = e^x + 1. The transformations involve shifting, stretching, or reflecting the parent graph of y = e^x.

a) The function's transformation from y = e^x involves shifting upward by 1 unit.

b) Domain: all real numbers.

c) Range: y > 1.

d) Horizontal asymptote: y = 1.

e) Y-intercept at x=0: f(0) = e⁰ + 1 = 1 + 1 = 2. Rounded to two decimal places: 2.00.

f) The graph resembles the standard exponential growth curve shifted upward, matching graph B, for example.

Overall, these solutions demonstrate the application of algebraic, exponential, and logarithmic principles in solving standard mathematical problems related to functions, their properties, and their transformations.

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