91Ó°ÊÓ

A quadratic function is a type of polynomial function that takes the form \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the exercise, the quadratic function is given as \( g(x) = 4x^2 \). This is a simple quadratic function where:

• The coefficient \( a \) is 4.
• There is no linear term (\( bx \)) or constant term (\( c \)).

Quadratic functions are useful to model scenarios where the effect grows or diminishes at an increasing rate. They are often graphed as parabolas. In this case, the function models a parabola opening upwards because the coefficient of \( x^2 \) is positive.

Substitution is a method used to evaluate functions by replacing variables with specific values. In our exercise, we substitute values into the function \( g(x) = 4x^2 \). The key is to carefully replace \( x \) with the given expression and perform the operations step by step. For instance:

• To find \( g\left(\frac{1}{a}\right) \), replace \( x \) with \( \frac{1}{a} \) to get \( 4\left(\frac{1}{a}\right)^2 = \frac{4}{a^2} \).
• For \( g(\sqrt{a}) \), substitute \( x \) with \( \sqrt{a} \), resulting in \( 4(\sqrt{a})^2 = 4a \).

Remember, substitution requires attention to detail to ensure each expression is correctly simplified. This skill becomes especially valuable when dealing with more complex functions and expressions.

The square root is a fundamental mathematical operation that finds a number whose square equals the given number. For example, the square root of 4 is 2 because \( 2^2 = 4 \). In our exercise, we see square roots used several times:

• When evaluating \( g(\sqrt{a}) \), where the square root of \( a \) is squared, simplifying to \( a \).
• When finding \( \sqrt{g(a)} \), which simplifies to \( \sqrt{4a^2} = 2a \).

Square roots "undo" squaring, so when applied to \( g(\sqrt{a}) \), the squared and square root operations cancel, leading to simplification. Square roots can become more complex when dealing with negative numbers, as real numbers do not have real square roots when negative under the square root symbol.

Inverse functions reverse the effect of the original function. If you have a function \( f(x) \), its inverse \( f^{-1}(x) \) will map the output back to the original input. However, only functions that are one-to-one (bijective) have inverses. In the context of the function \( g(x) = 4x^2 \), consider its inverse. The function \( g(x) \) does not have a straightforward inverse over all real numbers due to the squaring operation, which is not one-to-one for all real numbers, but it could be considered on restricted domains.For example, given \( g(x) = 4x^2 \), the inverse could be determined by solving \( y = 4x^2 \) for \( x \). After which, we can conclude an inverse involves a square root, but it is only valid for positive values of \( x \), reflecting the restriction needed for a true inverse. This demonstrates how useful but intricate inverses can be, requiring a deep understanding of both the function and its domain.