Evaluate The Function At The Indicated Values If An Answer I

Evaluate The Function At The Indicated Values If An Answer Is Undefi

Evaluate the function at the indicated values, and if an answer is undefined, enter UNDEFINED.

f(x) = 1 - 3x⁸ ; evaluate at x = 3, -3, a, -a, a - 1.

Similarly, evaluate h(x) = x² + 1 at x = 3, -3, a, -x, a - 3, x.

Also, evaluate f(x) = x² + 3x at x = 0, 3, -3, a, -x, and a = 4.

Compute f(x) = 5x² + 2 at x + 2, and simplify f(x + 2) and f(x) + f(2).

Evaluate f(x) = x + 9 at x², and (f(x))².

Find the domain and range of various functions, such as f(x) = 8x, f(x) = 5x² + 7, f(x) = x + 7 / (x²) - 1, f(x) = x⁴ / (x²) + x - 1, f(t) = t + 1, and interpret graphs of functions for domain, range, intervals of increase and decrease, and local extrema.

Paper For Above instruction

Evaluating functions at specific values, particularly when some inputs lead to undefined outputs, forms a core aspect of understanding basic algebraic functions and their behaviors. In this discussion, various functions and their evaluations are explored, along with the analysis of their domain and range, and graphical interpretations, to develop a comprehensive understanding of these concepts.

Starting with polynomial functions such as f(x) = 1 - 3x⁸, the evaluation at particular points like x=3 and x=-3 reveals the symmetric properties of even-powered polynomials. For example, substituting x=3 yields f(3)=1 - 3(3)⁸, which results in a large negative number due to the high even power, indicating that such functions grow rapidly in magnitude for larger values of |x|. Evaluations at x=-3, assuming similar substitutions, reveal symmetric outputs because even powers eliminate the negative sign. Evaluating the functions at variables such as a, -a, and a - 1 introduces parametric considerations, essential for understanding how functions behave over different inputs, including those dependent on variables.

Similarly, for h(x) = x² + 1, the evaluation at x=3 and x=-3 yields h(3)=9+1=10 and h(-3)=9+1=10, showcasing the even symmetry and the minimum value occurring at x=0. Evaluations at a, -x, a-3, and x demonstrate the function's domain over real numbers and reveal the parabolic shape typical of quadratic functions, with their range being [1, ∞). Such analysis illustrates the importance of understanding the structure of functions to interpret their graph shapes and value behaviors over various intervals.

Functions like f(x) = x² + 3x combine quadratic and linear behaviors, and their evaluations at different points help in determining their characteristics. For instance, at x=0, f(0)=0, the vertex of the parabola, and at x=3 or -3, the function values indicate how steeply the parabola ascends or descends. Incorporating parameters like a=4 introduces dynamism, allowing for a flexible exploration of function behaviors over various inputs.

Further, functions such as f(x) = 5x² + 2 are evaluated at x + 2 and x, leading to expressions like f(x+2) = 5(x+2)² + 2, which, when expanded, can be simplified to analyze shifts and transformations of the original parabola. The sum f(x) + f(2) involves evaluating f(2)=20+2=22, and combining with the general form to understand how composite functions behave. Such evaluations are fundamental in calculus and algebra for understanding transformations, shifts, and the behavior of functions over domains.

Function domain and range are critical components, often expressed in interval notation. For example, f(x)=8x has a domain of all real numbers (-∞, ∞), and its range is also all real numbers due to the linear nature. Conversely, quadratic functions like f(x) = 5x² + 7 have domains of (-∞, ∞) and ranges of [7, ∞), since the parabola opens upward with its vertex at the minimum point. Rational functions such as f(x) = (x+7)/(x² - 1) require an analysis of excluded values, namely where the denominator is zero, x=±1, to determine the domain as (-∞, -1) ∪ (-1, 1) ∪ (1, ∞).

Graphical analysis, as presented in various problems, involves estimating the domain and range, identifying intervals of increasing and decreasing behavior, as well as locating local maxima and minima. For example, a graph showing peaks and valleys can be examined to determine where the function ascends or descends, and the x-values at local extrema are significant. Recognizing these points enhances understanding of the function's overall behavior, crucial for calculus and analysis.

In conclusion, evaluating functions at specific points, understanding their domains and ranges, and interpreting their graphs form the foundation of algebraic analysis. These skills facilitate deeper insights into the behaviors of various functions, their transformations, and their extrema. Mastery over these concepts provides valuable tools for advanced mathematical studies, including calculus, where the analysis of function behaviors over different intervals is essential for understanding continuity, differentiability, and integrability.

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