Math 111 Final Exam Review: Graph Of Y=f(x) In Figure

Math 111 Final Exam Review1 Use The Graph Of Y Fx In Figure 1 To

Use the graph of y = f(x) in Figure 1 to answer the following. Approximate where necessary: (a) Evaluate f(−1). (b) Evaluate f(0). (c) Solve f(x) = 0. (d) Solve f(x) = −7. (e) Determine if f is even, odd, or neither from its graph. (f) State any local maximums or local minimums. (g) State the domain and range of f. (h) Over what interval(s) is the function increasing? (i) Over what interval(s) is the function decreasing? (j) Over what interval(s) is the function concave up? (k) Over what interval(s) is the function concave down? (l) Find the zeros of f. (m) Find a possible formula for this polynomial function.

Paper For Above instruction

The analysis of the graph of the function y = f(x) presented in Figure 1 enables us to explore various properties and behaviors of the function. Although the exact graph is not provided here, typical interpretations based on common graph behaviors can help clarify the process:

Evaluating f(−1) and f(0)

To approximate f(−1), locate −1 on the x-axis and identify the corresponding y-value on the graph. The y-value at this point approximates f(−1). Similarly, for f(0), find x = 0 and observe the y-value. If the graph crosses or approaches certain y-values around these x-values, those are the approximations.

Solving f(x) = 0 and f(x) = −7

To solve f(x) = 0, locate the points where the graph intersects the x-axis; these points' x-coordinates are the zeros of f. Approximating these points can be done visually or by reading off the graph if it’s clear. For f(x) = −7, identify the points on the graph where y = −7, either directly or via interpolation, to approximate the solutions for x.

Determining if f is even, odd, or neither

A function is even if its graph exhibits symmetry about the y-axis and odd if it exhibits symmetry about the origin. From the graph, observe whether f(x) = f(−x) holds for all x in the domain (indicating evenness), or whether f(−x) = −f(x) (indicating oddness). If neither symmetry is apparent, the function is neither even nor odd.

Identifying local extrema

Local maximums and minimums are points where the graph changes direction from increasing to decreasing or vice versa. Look for peaks (local maxima) and troughs (local minima) on the graph, noting their x-coordinates and the nature of the change in y-values around those points.

Domain and Range

The domain is the set of all x-values for which the graph exists, often all real numbers unless restricted by the graph's boundaries. The range is the set of all y-values attained by the function, which can be approximated by observing the highest and lowest points on the graph.

Intervals of increase and decrease

Identify where the graph moves upward as x increases (intervals of increase) and where it moves downward (intervals of decrease). These intervals can be approximated based on the slope of the graph over different x-intervals.

Concavity

Determine where the graph is concave up (shaped like a cup, curving upward) versus concave down (shaped like a cap, curving downward). Points where the concavity changes are inflection points — locate where the graph switches from convex to concave or vice versa.

Zeros and a possible polynomial formula

The zeros are the x-values where the graph crosses the x-axis. By identifying these zeros and their multiplicities (based on the behavior at the zeros — e.g., crossing or tangent), one can construct a potential polynomial function that fits the observed zeros and end behavior (long-term behavior).

Analysis of a Rational and Polynomial Function

Since specific figures are not provided, general techniques include identifying intercepts, asymptotic behavior, and end behavior. For polynomial functions, zeros and their multiplicities inform the degree and shape. For rational functions, asymptotes occur where the denominator approaches zero, and vertical and horizontal asymptotes help define the function's long-term behavior.

Transformations and Composition of Functions

Transformations of the absolute value function |x| include shifting, reflecting, stretching, and compressing. By examining the transformations, one can derive formulas such as f(x) = |x|, then apply the transformations in order for each case like −f(x), f(x + 1), 2f(x), and so forth, to find the simplified formulas describing the transformed graphs.

Function Operations and Their Values

Using the given functions, operations like addition, multiplication, and composition (g ◦ f or f ◦ g) are computed directly by substituting values from the tables and functions. These calculations reveal properties like intersecting points, combined behaviors, and the overall shape of the composite functions.

Applications: Exponential Decay and Investment Models

The exponential decay model V = (4/3)πr^3 describes the volume of a balloon with radius r, where r varies over time. The linear and exponential models of investment involve functions like Q(t) and T(t), whose behavior over time is interpreted in economic context. Calculations of specific values, inverse functions, and long-term estimates provide insights into decay rates and investment growths.

Transformations of Logarithms and Solving Equations

Expressing logarithms with exponents involves using properties such as log_b(a^c) = c * log_b(a). Solving exponential and logarithmic equations requires techniques like isolating the exponential and applying the inverse logarithm, along with algebraic manipulations to find exact and approximate solutions, followed by interpretation of the solutions.

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