91Ó°ÊÓ

Substitution is a fundamental concept in mathematics and is frequently used in function evaluation. It involves replacing the variable in a function with a specific value or expression. This allows us to calculate the function's output for that particular input.

In the provided exercise, substitution is used to find the values of the functions at a given point: f(x) at x=4 and g(x) at x=4.

For instance, with the function \( f(x) = -3x + 1 \), substituting x=4 yields \( f(4) = -3(4) + 1 \). This step-by-step substitution helps in simplifying and solving the function accurately.

Substitution is not just limited to numbers; it can also involve substituting expressions into functions, which is particularly useful in algebra.

Linear functions are those that produce a straight line when graphed and can be written in the form \( f(x) = mx + b \), where m and b are constants.

Here, \( m \) represents the slope of the line, which indicates the rate of change of the function. The constant \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

In our exercise, the function \( f(x) = -3x + 1 \) is linear.

Understanding the properties of linear functions helps us to evaluate and visualize them easily. When we input a value into a linear function, the output is dependent on the slope and the y-intercept, making the process straightforward.

Evaluating \( f(4) \) involves basic arithmetic, multiplying the slope by the given x-value and adding the y-intercept: \( f(4) = -3(4) + 1 = -12 + 1 = -11 \).

Quadratic functions are a type of polynomial function where the highest degree of the variable is 2. These functions can be expressed in the form \( g(x) = ax^2 + bx + c \).

In this specific problem, our quadratic function is simpler with only two terms: \( g(x) = x^2 + 2 \). The graph of a quadratic function is a parabola, which can open upwards or downwards, depending on the sign of the coefficient of \( x^2 \).

Here, the coefficient of \( x^2 \) is positive, so the parabola opens upwards. Evaluating \( g(x) \) at a specific point involves squaring the input value, multiplying by the coefficient (in this case, 1), and then adding the constant term.

For the problem, substituting x=4 into \( g(x) \) gives \( g(4) = 4^2 + 2 = 16 + 2 = 18 \). Quadratic functions often appear in various real-world applications, including physics and engineering.