How can we determine whether the functions fx = 4x^2, gx = 1/x, and hx = 32^x are linear, quadratic, or exponential?

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To classify the functions f(x) = 4x^2, g(x) = 1/x, and h(x) = 32^x as linear, quadratic, or exponential, we need to examine the form of each function and its characteristics.

1. Linear Functions

A linear function has the general form f(x) = mx + b, where m and b are constants. It produces a straight line when graphed. There are no exponentials or powers other than 1 applied to the variable.

2. Quadratic Functions

A quadratic function has the general form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a cannot be zero. The defining characteristic of the quadratic function is that it contains the variable x raised to the second power. It results in a parabola when graphed.

3. Exponential Functions

An exponential function has the general form f(x) = ab^x, where a is a constant, and b is the base raised to the power of x. The variable is in the exponent, causing rapid growth or decay when graphed.

Identifying Each Function

Now, let’s look at each function provided:

  • Function f(x) = 4x^2: This is a quadratic function because it has x raised to the power of 2. The coefficient 4 is simply a scaling factor that stretches the parabola vertically.
  • Function g(x) = 1/x: This represents a rational function, which is not linear, quadratic, or exponential. However, it can sometimes be analyzed similarly to linear or inverse functions. In this case, it may resemble a hyperbola when graphed.
  • Function h(x) = 32^x: This is an exponential function. The variable x is in the exponent, indicating rapid increase as x increases, which characterizes exponential growth.

Conclusion

In summary:

  • f(x) = 4x^2 is quadratic.
  • g(x) = 1/x is a rational function (neither linear, quadratic, nor exponential).
  • h(x) = 32^x is exponential.

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