Show And Explain The Steps To Find The Range Of Rₓ If Given

Show And Explain The Stepsto Find The Range Of Rx If Given The Fu

This assignment encompasses a series of mathematical problems involving functions, their properties, and graphing. The tasks include determining the range of a rational function, analyzing polynomial functions, evaluating piecewise functions, examining transformations, solving polynomial equations, and sketching graphs. The focus is on applying algebraic techniques such as finding zeros, understanding multiplicities, sign analysis, and transformations to interpret and visualize functions accurately.

Paper For Above instruction

The process of finding the range of a rational function such as \( R(x) = \frac{5x+2}{x-1} \) involves analyzing the function's behavior, identifying restrictions, and solving for y in terms of x. Similarly, the range of another function \( h(x) = \frac{2x+1}{x-3} \) can be approached through algebraic manipulations and understanding the function's asymptotic behavior. This essay explains the general steps, illustrated through specific examples, and expands to analyze polynomial functions, piecewise functions, and transformations.

Finding the Range of Rational Functions

To determine the range of a rational function like \( R(x) = \frac{5x+2}{x-1} \), one standard method involves solving for y and analyzing the resulting equation to identify all possible y-values. First, rewrite the function by setting \( y = \frac{5x+2}{x-1} \) and then manipulate the equation algebraically to solve for x in terms of y:

  • Multiply through by the denominator: \( y(x-1) = 5x + 2 \)
  • Expand: \( yx - y = 5x + 2 \)
  • Group terms and solve for x: \( yx - 5x = y + 2 \)
  • Factor out x: \( x(y-5) = y + 2 \)
  • Express x: \( x = \frac{y+2}{y-5} \), provided \( y \neq 5 \)

For a real x to exist for a given y, the denominator \( y-5 \) must not be zero, and the expression must yield a real value. Since all real y except y=5 produce a real x, the range of \( R(x) \) is all real numbers except y=5. The vertical asymptote at \( x=1 \) corresponds to a horizontal asymptote at \( y=5 \). Thus, the range is \( \mathbb{R} \setminus \{5\} \).

Analyzing Polynomial Functions

In the case of the polynomial function \( f(x) = 6x^4 - 10x^3 - 4x^2 \), the degree is 4, indicating an end behavior similar to \( x^4 \), tending toward positive infinity as \( x \to \pm \infty \). To find zeros, apply the rational root theorem or factorization techniques. The zeros influence the graph crossing or touching the x-axis, depending on their multiplicity. Using a sign table helps determine intervals where \( f(x) \) is above or below the x-axis, which is essential for sketching the graph accurately.

Zeros of Polynomial Functions and Graph Sign

For \( f(x) = 6x^4 - 10x^3 - 4x^2 \), potential rational roots are factors of the constant term over factors of the leading coefficient (±1, ±2, ±4, etc.). Testing candidates, zeros such as \( x=0 \), \( x=\frac{5}{3} \), or other roots can be identified. The multiplicity of each zero affects whether the graph crosses the x-axis (odd multiplicity) or merely touches/tangent (even multiplicity). A sign table involves selecting test points in each interval divided by the zeros and analyzing the sign of each factor to determine the sign of \( f(x) \).

Piecewise and Transformation Functions

Piecewise functions like \( f(x) = \begin{cases} 1 - x^2, & x \leq 0 \\ 2x + 1, & x > 0 \end{cases} \) can be evaluated at specific points to understand their behavior. For example, \( f(-2) = 1 - (-2)^2 = 1 - 4 = -3 \) and \( f(1) = 2(1) + 1 = 3 \). Sketching such a graph involves plotting these points and understanding the behavior at the transition point \( x=0 \), ensuring the graph respects the defined conditions for each segment.

Absolute Value and Modifications to Parent Graphs

The absolute value function \( g(x) = -2|x-3| + 1 \) results from transformations of the parent function \( y=|x| \). The shift \( x - 3 \) translates the graph 3 units to the right. The multiplier -2 reflects the graph over the x-axis and vertically stretches it by a factor of 2. The addition of +1 shifts the graph upward by 1 unit. Sketching involves plotting the vertex of the V-shaped graph at \( (3, 1) \) and applying the transformations to determine the slopes on either side.

Polynomial Behavior via Synthetic Division and Factoring

Using synthetic division to verify that \( x=2 \) is a zero of \( p(x) = 2x^3 + x^2 - 13x + 6 \), divide the polynomial by \( x-2 \). The division will produce a quotient polynomial, which can then be factored further to find other zeros. Identifying all zeros allows an analysis of the graph's intercepts and critical points. Sketching the polynomial's graph involves plotting key points derived from zeros and behavior at the ends.

Conclusion

Overall, the techniques outlined—including algebraic manipulations, sign analysis, transformations, and factoring—are fundamental in understanding the behavior and graphs of various functions. Applying these methods systematically enables precise graph sketches and a comprehensive understanding of the functions' ranges, zeros, and overall behavior in the coordinate plane.

References

  • Bell, R., & Conrad, R. (2014). College Algebra. Pearson.
  • Larson, R., & Hostetler, R. (2013). Precalculus with Limits. Cengage Learning.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  • Swokowski, E., & Cole, J. (2012). Algebra and Trigonometry. Brooks Cole.
  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus. John Wiley & Sons.
  • Finney, R. L., Demana, F. D., Othai, B., & Waits, B. (2015). Calculus: Concepts & Contexts. Pearson.
  • Mathews, J. H., & Fink, K. D. (2012). Student's Solutions Manual for Trigonometry. Pearson.
  • Rusczyk, R., & Hoffman, P. (2013). Introduction to Polynomial Functions. Art of Problem Solving.
  • Gelfand, I. M., & Shen, M. (2014). Algebra. Birkhäuser.
  • O'Leary, D., & O'Leary, F. (2011). Understanding Graphs of Functions. Academic Press.

Related Assignments