Problem 78 - College Algebra

Question

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. $$g(f(-3))$$

Step-by-Step Solution

Answer

The value of the composite function \( g(f(-3)) \) is 62.

Step 1: Evaluate the inside function

We start by evaluating the function \( f(x) = 2x^2 + 1 \) at \( x = -3 \). First, calculate \( (-3)^2 = 9 \). Then multiply by 2 to get \( 2 \times 9 = 18 \). Finally, add 1 to get \( f(-3) = 18 + 1 = 19 \).

Step 2: Substitute into the outside function

Now that we have \( f(-3) = 19 \), we substitute this value into the function \( g(x) = 3x + 5 \). We replace \( x \) with 19 in \( g(x) \): \( g(19) = 3 \times 19 + 5 \).

Step 3: Evaluate the outside function

Calculate \( 3 \times 19 = 57 \). Then add 5 to this result to get \( 57 + 5 = 62 \). So, \( g(19) = 62 \).

Key Concepts

Function Evaluation

Function evaluation is a process where we calculate the value of a function for a specific input. This is an essential part of solving mathematical problems, especially when dealing with different types of functions, such as polynomial functions or composite functions.

To start evaluating a function, use one of its input values, often represented by the variable \( x \). For example, consider the function \( f(x) = 2x^2 + 1 \). To evaluate \( f \) at \( x = -3 \), substitute \( -3 \) into the function. Here's how it's done:

• First, solve for \((-3)^2\), which is \(9\).
• Multiply \(9\) by \(2\) to get \(18\).
• Add \(1\) to \(18\), resulting in \(19\).

Hence, \( f(-3) = 19 \). This step is crucial because it gives you a specific point in the function's range that will be used in further calculations.

Polynomial Functions

Polynomial functions are a fundamental class of functions in mathematics, expressed in the form \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants, and \( n \) indicates the degree of the polynomial.

These functions are versatile and can take many shapes, such as parabolas, which are common in quadratic polynomials. The function \( f(x) = 2x^2 + 1 \) is a quadratic polynomial function. This means the highest power of \( x \) is 2, giving it a parabolic shape.
By analyzing the components of a polynomial function, you can predict its general behavior:

• The coefficient of \( x^2 \) (here \(2\)), determines how "wide" or "narrow" the parabola is.
• The constant term \( (1) \) affects where the graph intersects the \( y \)-axis.

Understanding these attributes helps in evaluating and graphing polynomial functions, making it easier to visualize their behavior and solve related mathematical tasks.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. This concept is written as \((g \circ f)(x)\), meaning \(g(f(x))\). It requires performing a sequence of operations: first applying the inside function, then the outside function.

Our exercise uses function composition to find \(g(f(-3))\). Start by evaluating \( f(-3) \), which we calculated as \(19\). This result becomes the input for the function \( g(x) \). We substitute \(19\) into \( g(x) = 3x + 5 \):

• Calculate \(3 \times 19 = 57\).
• Add \(5\) to \(57\), giving \(62\).

Thus, \(g(f(-3)) = 62\).
Function composition is useful because it allows us to create complex functions from simpler ones, streamlining problem-solving processes in calculus and advanced algebra.