Chapter 4: Problem 26
\(g(x)=x^3+x+3\)
Short Answer
Expert verified
The given function \(g(x)=x^3+x+3\) is a cubic function that increases as \(x\) increases. It passes through points \((-1, 1), (0, 3), (1, 5)\).
Step by step solution
01
Identify the type of function
Analyze the given function. This function \(g(x)=x^3+x+3\) appears to be a cubic function since the highest power of variable \(x\) is 3.
02
Determine the behavior of the function at different points
Any x-values could be used to see the behavior of the function at different points. For example, for \(x = -1, 0, 1\), plug those values into the expression and find the function values: \(g(-1)\) equals to \((-1)^3+(-1)+3=1\), \(g(0)\) equals to \(0^3+0+3=3\), \(g(1)\) equals to \(1^3+1+3=5\). So, the function passes through points \((-1, 1), (0, 3), (1, 5)\).
03
Sketch the function
Based on identified points \((-1, 1), (0, 3), (1, 5)\) and the nature of a cubic function, one can sketch the function. It would increase as \(x\) increases, passing through those three points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole-number exponents. In the case of the function given, \(g(x) = x^3 + x + 3\), it is a cubic polynomial function. This is because the highest power of the variable \(x\) is 3.
Such functions can have various forms:
Such functions can have various forms:
- They are generally expressed as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(n\) is a non-negative integer.
- Each term includes a coefficient and a power of \(x\), and coefficients can be any real number.
- The degree of the polynomial is determined by the highest exponent present, which dictates its general shape.
Graphing Functions
Graphing functions allows us to visually interpret the behavior of polynomial functions, like our cubic function \(g(x) = x^3 + x + 3\). Plotting key points derived from the equation aids in creating a sketch.
To graph, first find relevant points by choosing several x-values. For our function:
For cubic functions, observe symmetry and transitory behavior often reflected around critical points. Visualizing the curve’s increase signifies how \(g(x)\) continues past these plotted points. As \(x\) grows in the negative or positive direction, a cubic curve typically continues to increase or decrease, illustrating its characteristic 'S' shape.
To graph, first find relevant points by choosing several x-values. For our function:
- At \(x = -1\), \(g(-1)\) calculates to 1;
- \(x = 0\), \(g(0)\) calculates to 3;
- And \(x = 1\), \(g(1)\) calculates to 5.
For cubic functions, observe symmetry and transitory behavior often reflected around critical points. Visualizing the curve’s increase signifies how \(g(x)\) continues past these plotted points. As \(x\) grows in the negative or positive direction, a cubic curve typically continues to increase or decrease, illustrating its characteristic 'S' shape.
Function Analysis
Function analysis is essential in comprehending the complete behavior of polynomial expressions beyond just plotting it. Analysis of the function \(g(x) = x^3 + x + 3\) entails understanding its derivatives, intercepts, and overall behavior.
1. **Intercepts:** - Finding the roots, solving \(x^3 + x + 3 = 0\), determines where the graph intersects the x-axis. - The y-intercept is immediately recognizable at \(g(0) = 3\). This point is \((0,3)\) on the graph.2. **Derivatives:** - The first derivative \(g'(x) = 3x^2 + 1\) provides insights on the increasing or decreasing nature. - The second derivative \(g''(x) = 6x\) informs us about the curvature – concave up or down.3. **Behavior at extremes:** - As \(x\) approaches positive or negative infinity, \(x^3\) becomes dominant, influencing the rise and fall of the function.Through these methods, deeper insights into the function's critical points and curvature, as well as asymptotic behavior, can be achieved. This deeper analysis aids in completely mastering the essence of cubic functions.
1. **Intercepts:** - Finding the roots, solving \(x^3 + x + 3 = 0\), determines where the graph intersects the x-axis. - The y-intercept is immediately recognizable at \(g(0) = 3\). This point is \((0,3)\) on the graph.2. **Derivatives:** - The first derivative \(g'(x) = 3x^2 + 1\) provides insights on the increasing or decreasing nature. - The second derivative \(g''(x) = 6x\) informs us about the curvature – concave up or down.3. **Behavior at extremes:** - As \(x\) approaches positive or negative infinity, \(x^3\) becomes dominant, influencing the rise and fall of the function.Through these methods, deeper insights into the function's critical points and curvature, as well as asymptotic behavior, can be achieved. This deeper analysis aids in completely mastering the essence of cubic functions.