Question

\(f(x)=x^5-2 x+3, g(x)=3 f(x)\)

Short answer

The function \( g(x) \) is \( g(x) = 3x^5 - 6x + 9 \).

Step-by-step solution

Identify the Functions

Start by identifying the functions given in the problem. In this case, the function \( f(x) = x^5 - 2x + 3 \) has been provided.

Express \( g(x) \) in Terms of \( f(x) \)

The problem states that \( g(x) = 3f(x) \). This means that every element of \( f(x) \) should be multiplied by 3.

Find the Function \( g(x) \)

To find the function \( g(x) \), substitute \( f(x) = x^5 - 2x + 3 \) into the expression for \( g(x) \). This gives \( g(x) = 3(x^5 - 2x + 3) \). Then, distribute the 3 to all the terms inside the bracket. So, \( g(x) = 3x^5 - 6x + 9 \).

Key concepts

Understanding Function Operations

Function operations include adding, subtracting, multiplying, and even dividing functions. In our exercise, we see multiplication, where the function \(g(x)\) is defined as a multiscalar operation on \(f(x)\).
To perform these operations, follow these steps:

• Identify the operation needed (e.g., adding, subtracting, multiplying, or dividing the functions).
• Apply the operation step-by-step to combine, separate, or scalar multiply each term of the functions as needed.

In the example, \(g(x) = 3f(x)\) represents a scalar multiplication. Here, each term in \(f(x)\) is multiplied by a constant to form the new function \(g(x) = 3(x^5 - 2x + 3)\). Hence, function operations allow creating new, often more complex functions by manipulating existing ones.

Exploring Function Composition

Function composition involves using the output of one function as the input to another function, written as \((f \circ g)(x)\) or simply \(f(g(x))\).
While our problem doesn't directly involve function composition, understanding it is crucial in advanced math topics.
Here's how you perform function composition:

• Given two functions, let's say \(f(x)\) and \(g(x)\), evaluate \(g(x)\) first.
• Substitute the result of \(g(x)\) into \(f(x)\) to find the composite function \(f(g(x))\).

This operation can be particularly powerful when multiple layers of functions are involved. It allows for building highly customized functions, which can model real-world phenomena in a structured manner.

How to Approach Transformation of Functions

Transformation of functions involves shifting, stretching, compressing, or flipping a function graph. These changes do not alter the nature of the function but adjust its graphical representation.
Transformations include:

• Vertical shifts - moving the graph up or down.
• Horizontal shifts - moving the graph left or right.
• Stretching and compressing - altering the graph's width or height.
• Reflecting - flipping the graph over an axis.

In our exercise's context, multiplying \(f(x)\) by the scalar 3 represents a vertical stretch. This operation increases the distance between each y-value and the x-axis, thereby scaling the function vertically by a factor of 3. Understanding transformations helps in quickly visualizing how alterations to a function's equation affect its graph.