Two Functions And Are Defined In The Figure Below

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Functions, the fundamental building blocks of mathematics and computer science, serve as powerful tools for describing relationships between variables and performing specific operations. Understanding the properties and interactions of different types of functions is crucial for problem-solving and mathematical modeling. In this exploration, we will delve into two distinct functions, denoted as ( f(x) ) and ( g(x) ), carefully examining their definitions, behaviors, and relationships.

Understanding the Definitions of f(x) and g(x)

To begin, it's essential to establish clear definitions for the functions ( f(x) ) and ( g(x) ). The definition of a function specifies the rule or procedure that maps each input value (x) to a unique output value.

• Definition of ( f(x) ):

  The function ( f(x) ) is defined as:

  $ f(x) = x^2 + 3x - 5 $

  This is a quadratic function, characterized by the highest power of ( x ) being 2. Quadratic functions are known for their parabolic shape when graphed, and their behavior is largely determined by the coefficients of the terms ( x^2 ), ( x ), and the constant term.

• Definition of ( g(x) ):

  The function ( g(x) ) is defined as:

  $ g(x) = 2\sin(x) + 1 $

  This is a trigonometric function, specifically a sine function. The sine function is periodic, oscillating between -1 and 1, and it is commonly used to model periodic phenomena such as waves and oscillations. The ( 2 ) in front of ( \sin(x) ) scales the amplitude of the sine wave, while the ( +1 ) shifts the entire function vertically.

Domain and Range of f(x) and g(x)

The domain of a function is the set of all possible input values (x) for which the function is defined. The range is the set of all possible output values that the function can produce.

• Domain and Range of ( f(x) ):

  • Domain: Since ( f(x) = x^2 + 3x - 5 ) is a polynomial function, it is defined for all real numbers. Therefore, the domain of ( f(x) ) is ( (-\infty, \infty) ).

  • Range: To find the range, we need to determine the minimum value of the quadratic function. The vertex of the parabola ( f(x) = x^2 + 3x - 5 ) occurs at ( x = -\frac{b}{2a} ), where ( a = 1 ) and ( b = 3 ). Thus, ( x = -\frac{3}{2} ). Plugging this value into ( f(x) ) gives:

    $ f\left(-\frac{3}{2}\right) = \left(-\frac{3}{2}\right)^2 + 3\left(-\frac{3}{2}\right) - 5 = \frac{9}{4} - \frac{9}{2} - 5 = \frac{9 - 18 - 20}{4} = -\frac{29}{4} $

    Since the parabola opens upwards (as the coefficient of ( x^2 ) is positive), the minimum value of ( f(x) ) is ( -\frac{29}{4} ). Therefore, the range of ( f(x) ) is ( \left[-\frac{29}{4}, \infty\right) ).

• Domain and Range of ( g(x) ):

  • Domain: The sine function is defined for all real numbers. Therefore, the domain of ( g(x) = 2\sin(x) + 1 ) is ( (-\infty, \infty) ).
  • Range: The sine function oscillates between -1 and 1, i.e., ( -1 \leq \sin(x) \leq 1 ). Multiplying by 2, we get ( -2 \leq 2\sin(x) \leq 2 ). Adding 1 to all sides, we have ( -1 \leq 2\sin(x) + 1 \leq 3 ). Thus, the range of ( g(x) ) is ( [-1, 3] ).

Analyzing the Graphs of f(x) and g(x)

Visualizing functions through their graphs provides valuable insights into their behavior.

• Graph of ( f(x) = x^2 + 3x - 5 ):

  As mentioned earlier, ( f(x) ) is a quadratic function, and its graph is a parabola. The parabola opens upwards because the coefficient of ( x^2 ) is positive. The vertex of the parabola is at ( \left(-\frac{3}{2}, -\frac{29}{4}\right) ), and the parabola is symmetric about the vertical line ( x = -\frac{3}{2} ).

  The graph of ( f(x) ) intersects the x-axis at the points where ( f(x) = 0 ). To find these points, we solve the quadratic equation ( x^2 + 3x - 5 = 0 ). Using the quadratic formula:

  $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{3^2 - 4(1)(-5)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 20}}{2} = \frac{-3 \pm \sqrt{29}}{2} $

  So, the x-intercepts are approximately ( x \approx 1.19 ) and ( x \approx -4.19 ).

• Graph of ( g(x) = 2\sin(x) + 1 ):

  The graph of ( g(x) ) is a sine wave. The amplitude of the wave is 2, and the entire wave is shifted up by 1 unit. The graph oscillates between -1 and 3. The period of the sine wave is ( 2\pi ), which means the pattern repeats every ( 2\pi ) units along the x-axis.

  The graph of ( g(x) ) intersects the x-axis at the points where ( g(x) = 0 ). To find these points, we solve the equation ( 2\sin(x) + 1 = 0 ), which simplifies to ( \sin(x) = -\frac{1}{2} ). The solutions for ( x ) in the interval ( [0, 2\pi] ) are ( x = \frac{7\pi}{6} ) and ( x = \frac{11\pi}{6} ). Since the sine function is periodic, the general solutions are ( x = \frac{7\pi}{6} + 2n\pi ) and ( x = \frac{11\pi}{6} + 2n\pi ), where ( n ) is an integer.

Transformations of Functions

Understanding transformations allows us to manipulate and analyze functions more effectively. Common transformations include translations, reflections, and scaling.

• Transformations of ( f(x) ):

  The function ( f(x) = x^2 + 3x - 5 ) can be seen as a transformation of the basic quadratic function ( y = x^2 ). The transformations include:

  1. Horizontal Translation: The term ( 3x ) in ( x^2 + 3x ) suggests a horizontal translation. To find the exact translation, we complete the square:

     $ f(x) = x^2 + 3x - 5 = \left(x + \frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 - 5 = \left(x + \frac{3}{2}\right)^2 - \frac{9}{4} - 5 = \left(x + \frac{3}{2}\right)^2 - \frac{29}{4} $

     This shows that the graph of ( f(x) ) is the graph of ( y = x^2 ) translated ( \frac{3}{2} ) units to the left and ( \frac{29}{4} ) units down.

  2. Vertical Translation: As seen in the completed square form, the graph is translated ( \frac{29}{4} ) units down.

• Transformations of ( g(x) ):

  The function ( g(x) = 2\sin(x) + 1 ) can be seen as a transformation of the basic sine function ( y = \sin(x) ). The transformations include:

  1. Vertical Stretch: The coefficient 2 in ( 2\sin(x) ) stretches the sine function vertically by a factor of 2, effectively doubling the amplitude.
  2. Vertical Translation: The term ( +1 ) shifts the entire graph of the sine function upward by 1 unit.

Composite Functions: Combining f(x) and g(x)

A composite function is a function that is formed by applying one function to the result of another. For example, ( f(g(x)) ) means that we first apply the function ( g(x) ) to ( x ), and then apply the function ( f(x) ) to the result.

• ( f(g(x)) ):

  To find ( f(g(x)) ), we substitute ( g(x) ) into ( f(x) ):

  $ f(g(x)) = f(2\sin(x) + 1) = (2\sin(x) + 1)^2 + 3(2\sin(x) + 1) - 5 $

  Expanding and simplifying:

  $ f(g(x)) = (4\sin^2(x) + 4\sin(x) + 1) + (6\sin(x) + 3) - 5 = 4\sin^2(x) + 10\sin(x) - 1 $

  So, ( f(g(x)) = 4\sin^2(x) + 10\sin(x) - 1 ).

• ( g(f(x)) ):

  To find ( g(f(x)) ), we substitute ( f(x) ) into ( g(x) ):

  $ g(f(x)) = g(x^2 + 3x - 5) = 2\sin(x^2 + 3x - 5) + 1 $

  So, ( g(f(x)) = 2\sin(x^2 + 3x - 5) + 1 ).

The composite functions ( f(g(x)) ) and ( g(f(x)) ) are quite different, illustrating that the order in which functions are composed matters.

Derivatives of f(x) and g(x)

The derivative of a function measures the rate at which the function's output changes with respect to its input. It provides valuable information about the function's slope and behavior.

• Derivative of ( f(x) ):

  The derivative of ( f(x) = x^2 + 3x - 5 ) is found using the power rule:

  $ f'(x) = \frac{d}{dx}(x^2 + 3x - 5) = 2x + 3 $

• Derivative of ( g(x) ):

  The derivative of ( g(x) = 2\sin(x) + 1 ) is found using the derivative of the sine function:

  $ g'(x) = \frac{d}{dx}(2\sin(x) + 1) = 2\cos(x) $

Integrals of f(x) and g(x)

The integral of a function represents the area under the curve of the function. It is the reverse operation of differentiation.

• Integral of ( f(x) ):

  The integral of ( f(x) = x^2 + 3x - 5 ) is found using the power rule for integration:

  $ \int f(x) , dx = \int (x^2 + 3x - 5) , dx = \frac{1}{3}x^3 + \frac{3}{2}x^2 - 5x + C $

  where ( C ) is the constant of integration.

• Integral of ( g(x) ):

  The integral of ( g(x) = 2\sin(x) + 1 ) is found using the integral of the sine function:

  $ \int g(x) , dx = \int (2\sin(x) + 1) , dx = -2\cos(x) + x + C $

  where ( C ) is the constant of integration.

Applications of f(x) and g(x)

Functions like ( f(x) ) and ( g(x) ) have numerous applications in various fields:

• Quadratic Functions ( f(x) ):
  • Physics: Modeling projectile motion, such as the trajectory of a ball thrown in the air.
  • Engineering: Designing parabolic reflectors used in satellite dishes and car headlights.
  • Economics: Describing cost and revenue curves in business models.
• Trigonometric Functions ( g(x) ):
  • Physics: Modeling wave phenomena such as sound waves, light waves, and oscillations in mechanical systems.
  • Engineering: Analyzing alternating current (AC) circuits and signal processing.
  • Mathematics: Analyzing periodic phenomena and oscillations.

Solving Equations Involving f(x) and g(x)

Solving equations involving ( f(x) ) and ( g(x) ) can provide insights into their relationships and behaviors.

• Solving ( f(x) = 0 ):

  We already found the solutions for ( f(x) = 0 ) earlier using the quadratic formula:

  $ x = \frac{-3 \pm \sqrt{29}}{2} $

  These are the x-intercepts of the graph of ( f(x) ).

• Solving ( g(x) = 0 ):

  We also found the solutions for ( g(x) = 0 ) earlier:

  $ x = \frac{7\pi}{6} + 2n\pi \quad \text{and} \quad x = \frac{11\pi}{6} + 2n\pi $

  where ( n ) is an integer. These are the x-intercepts of the graph of ( g(x) ).

• Solving ( f(x) = g(x) ):

  To find the points where ( f(x) = g(x) ), we set the two functions equal to each other:

  $ x^2 + 3x - 5 = 2\sin(x) + 1 $

  $ x^2 + 3x - 6 = 2\sin(x) $

  This equation is more complex and generally requires numerical methods to solve. The solutions would represent the x-values where the graphs of ( f(x) ) and ( g(x) ) intersect.

Key Properties and Characteristics

• ( f(x) = x^2 + 3x - 5 )
  • Domain: ( (-\infty, \infty) )
  • Range: ( \left[-\frac{29}{4}, \infty\right) )
  • Vertex: ( \left(-\frac{3}{2}, -\frac{29}{4}\right) )
  • X-intercepts: ( x = \frac{-3 \pm \sqrt{29}}{2} )
  • Parabolic shape, opens upwards
• ( g(x) = 2\sin(x) + 1 )
  • Domain: ( (-\infty, \infty) )
  • Range: ( [-1, 3] )
  • Amplitude: 2
  • Period: ( 2\pi )
  • X-intercepts: ( x = \frac{7\pi}{6} + 2n\pi ) and ( x = \frac{11\pi}{6} + 2n\pi ), where ( n ) is an integer
  • Oscillating sine wave

Examples and Numerical Values

To further illustrate the behavior of these functions, let's consider some numerical examples:

• Values of ( f(x) ):

  • ( f(0) = 0^2 + 3(0) - 5 = -5 )
  • ( f(1) = 1^2 + 3(1) - 5 = -1 )
  • ( f(-1) = (-1)^2 + 3(-1) - 5 = 1 - 3 - 5 = -7 )
  • ( f(2) = 2^2 + 3(2) - 5 = 4 + 6 - 5 = 5 )

• Values of ( g(x) ):

  • ( g(0) = 2\sin(0) + 1 = 2(0) + 1 = 1 )
  • ( g\left(\frac{\pi}{2}\right) = 2\sin\left(\frac{\pi}{2}\right) + 1 = 2(1) + 1 = 3 )
  • ( g(\pi) = 2\sin(\pi) + 1 = 2(0) + 1 = 1 )
  • ( g\left(\frac{3\pi}{2}\right) = 2\sin\left(\frac{3\pi}{2}\right) + 1 = 2(-1) + 1 = -1 )

Conclusion

In summary, ( f(x) = x^2 + 3x - 5 ) is a quadratic function represented by a parabola, while ( g(x) = 2\sin(x) + 1 ) is a trigonometric function represented by a sine wave. Understanding their definitions, domains, ranges, transformations, derivatives, integrals, and applications provides a comprehensive overview of their behavior and significance in mathematics and various fields. Exploring their composite functions and solving equations involving these functions further enriches our understanding of their interplay. These fundamental concepts are essential for further studies in mathematics, physics, engineering, and other quantitative disciplines.