91Ó°ÊÓ

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in a single variable, like the ones in our exercise, takes the form:

• Constant term
• Linear term
• Quadratic term
• Cubic term

The degree of the polynomial is determined by the highest power of the variable in the expression. For instance, the polynomial \( h(x) = 4x^2 + 4x + 7 \) is a quadratic polynomial because its highest exponent is 2. Similarly, for \( f(x) = 3x + 5 \), we see a linear polynomial as the highest power is 1.
Each polynomial serves specific roles based on its structure. Quadratic polynomials are expressed as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Understanding these basic elements helps in analyzing and comparing polynomial functions.

Transformation of functions involves changing a function's appearance by applying modifications. These transformations can include shifts, stretches, compressions, or reflections. In the context of our exercise, function composition is a key transformation method. Here, we apply more than one function sequentially to achieve the desired outcome.
To find a function \( f \) resulting in \( h \) through composition with \( g \), i.e., \( f \circ g = h \), we determined that \( h(x) = 4x^2 + 4x + 7 \) almost fits the squared form \((2x+1)^2 = 4x^2 + 4x + 1 \). By adding 6, we fine-tuned it to match \( h(x) \). Hence, \( f(u) = u^2 + 6 \) was formed where \( u = g(x) \).
Similarly, transformations were used in part (b) by engineering \( g \) to fit \( f \circ g \) for \( h(x) \), showing that transforming functions can significantly alter their structure and application.

Verification is a crucial step in mathematical problem-solving to ensure the correctness of solutions. After deriving a function, it’s essential to check its validity by substituting back into the composition equation.
For instance, when we found \( f(u) = u^2 + 6 \) for the function composition in part (a), verifying involved plugging \( g(x) = 2x + 1 \) into the formula for \( f \) and simplifying. We confirmed that \( f(g(x)) = 4x^2 + 4x + 7 \), identical to \( h(x) \).
In part (b), verifying \( g(x) = x^2 + x - 1 \) was achieved by replacing \( g(x) \) in \( f \) and checking if it equates to \( h(x) = 3x^2 + 3x + 2 \). Verifying solutions by retracing steps affirms accuracy and strengthens understanding in mathematical exercises.