Evaluating Combined Functions A Comprehensive Guide

This article provides a comprehensive guide on how to evaluate combined functions, covering various operations such as addition, subtraction, multiplication, and division. We will also explore the evaluation of composite functions with constant values. Understanding these operations is crucial for mastering advanced mathematical concepts and solving complex problems. This guide aims to provide clarity and step-by-step explanations to help you confidently tackle function evaluations.

Understanding Function Operations

In the realm of mathematics, functions are fundamental building blocks, and understanding how to manipulate them is essential. Operations on functions, such as addition, subtraction, multiplication, and division, enable us to create new functions and analyze their behavior. This section aims to provide a thorough understanding of these operations, offering clarity and detailed explanations to help you confidently tackle function evaluations.

1. Addition of Functions: $(f+a)(x)$

Function addition is a straightforward operation where we combine two functions, $f(x)$ and $a(x)$, by adding their corresponding values for each input $x$. To find $(f+a)(x)$, you simply add the expressions of the two functions together. This operation results in a new function that represents the sum of the original functions.

When dealing with function addition, the key is to identify like terms and combine them. For example, if $f(x) = x^2 + 2x + 1$ and $a(x) = 3x^2 - x + 2$, then $(f+a)(x)$ would be found by adding the corresponding terms: $(x^2 + 3x^2) + (2x - x) + (1 + 2)$, which simplifies to $4x^2 + x + 3$. The resulting function, $4x^2 + x + 3$, represents the sum of the two original functions.

Understanding the domain of the resulting function is also crucial in function addition. The domain of $(f+a)(x)$ is the intersection of the domains of $f(x)$ and $a(x)$. This means that the values of $x$ for which $(f+a)(x)$ is defined must be within the domains of both original functions. For instance, if $f(x)$ is defined for all real numbers and $a(x)$ is defined for $x > 0$, then the domain of $(f+a)(x)$ is $x > 0$. This consideration ensures that the sum is only computed for valid inputs in both functions.

The graphical interpretation of function addition involves adding the y-values of the two functions at each x-value. If you were to plot $f(x)$ and $a(x)$ on the same coordinate plane, the graph of $(f+a)(x)$ would be obtained by vertically adding the corresponding points on the two original graphs. This visualization provides an intuitive understanding of how the combined function behaves based on the individual functions.

2. Subtraction of Functions: $(f-a)(x)$

Function subtraction is similar to addition but involves subtracting the values of one function from another. To find $(f-a)(x)$, you subtract the expression of $a(x)$ from the expression of $f(x)$. This operation is particularly sensitive to the order of subtraction, as $(f-a)(x)$ is generally different from $(a-f)(x)$.

When performing function subtraction, attention to the signs is crucial. If $f(x) = 2x^3 - 4x + 5$ and $a(x) = x^3 + 2x - 1$, then $(f-a)(x)$ is found by subtracting $a(x)$ from $f(x)$: $(2x^3 - 4x + 5) - (x^3 + 2x - 1)$. Distributing the negative sign and combining like terms yields $2x^3 - 4x + 5 - x^3 - 2x + 1$, which simplifies to $x^3 - 6x + 6$. The resulting function, $x^3 - 6x + 6$, represents the difference between the two original functions.

As with addition, the domain of the resulting function is a critical consideration in subtraction. The domain of $(f-a)(x)$ is the intersection of the domains of $f(x)$ and $a(x)$. This ensures that the subtraction is performed only for values of $x$ for which both functions are defined. For example, if $f(x)$ is defined for all real numbers and $a(x)$ is defined for $x eq 2$, then the domain of $(f-a)(x)$ is all real numbers except $2$.

The graphical interpretation of function subtraction involves subtracting the y-values of $a(x)$ from the y-values of $f(x)$ at each x-value. If you plot $f(x)$ and $a(x)$ on the same coordinate plane, the graph of $(f-a)(x)$ would be obtained by vertically subtracting the points on the graph of $a(x)$ from the corresponding points on the graph of $f(x)$. This visualization aids in understanding how the difference between the functions varies across different values of $x$.

3. Multiplication of Functions: $(f imes a)(x)$

Function multiplication involves multiplying the expressions of two functions, $f(x)$ and $a(x)$, together. To find $(f imes a)(x)$, you multiply the entire expression of $f(x)$ by the entire expression of $a(x)$. This operation often involves the distributive property and can result in a more complex function.

When performing function multiplication, it is essential to apply the distributive property correctly. For instance, if $f(x) = x + 3$ and $a(x) = x^2 - 2x + 1$, then $(f imes a)(x)$ is found by multiplying the two expressions: $(x + 3)(x^2 - 2x + 1)$. Distributing each term in the first expression across the terms in the second expression yields $x(x^2 - 2x + 1) + 3(x^2 - 2x + 1)$, which expands to $x^3 - 2x^2 + x + 3x^2 - 6x + 3$. Combining like terms simplifies the expression to $x^3 + x^2 - 5x + 3$. The resulting function, $x^3 + x^2 - 5x + 3$, represents the product of the two original functions.

The domain of the resulting function in multiplication is, again, the intersection of the domains of the individual functions. The domain of $(f imes a)(x)$ consists of all values of $x$ for which both $f(x)$ and $a(x)$ are defined. This ensures that the product is computed using valid inputs for both functions. For example, if $f(x)$ is defined for all real numbers and $a(x)$ is defined for $x eq 0$, then the domain of $(f imes a)(x)$ is all real numbers except $0$.

Graphically, function multiplication can be less intuitive than addition or subtraction. The y-values of the resulting function $(f imes a)(x)$ are the product of the y-values of $f(x)$ and $a(x)$ at each x-value. This means that the graph of $(f imes a)(x)$ will show how the product of the functions behaves, which can lead to amplifications, diminutions, or sign changes depending on the values of the original functions. Visualizing function multiplication often requires careful consideration of the behavior of each function and their combined effect.

4. Division of Functions: $rac{f(x)}{g(x)}$

Function division involves dividing one function, $f(x)$, by another function, $g(x)$. To find $rac{f(x)}{g(x)}$, you divide the expression of $f(x)$ by the expression of $g(x)$. This operation introduces a critical constraint: the denominator, $g(x)$, cannot be equal to zero, as division by zero is undefined.

When performing function division, the expression is written as a fraction, with $f(x)$ in the numerator and $g(x)$ in the denominator. For instance, if $f(x) = x^2 - 4$ and $g(x) = x + 2$, then $rac{f(x)}{g(x)}$ is expressed as $rac{x^2 - 4}{x + 2}$. Simplifying the expression often involves factoring and canceling common factors. In this case, $x^2 - 4$ can be factored as $(x + 2)(x - 2)$, so the expression becomes $rac{(x + 2)(x - 2)}{x + 2}$. Canceling the common factor of $(x + 2)$ simplifies the expression to $x - 2$. However, it is crucial to remember that the original function is undefined when $x = -2$, so this value must be excluded from the domain.

The domain of the resulting function in division is the intersection of the domains of $f(x)$ and $g(x)$, excluding any values of $x$ for which $g(x) = 0$. This is a critical consideration in function division. For example, if $f(x)$ is defined for all real numbers and $g(x)$ is defined for all real numbers except $x = -2$, and $g(x) = 0$ when $x = -2$, then the domain of $rac{f(x)}{g(x)}$ is all real numbers except $-2$. This ensures that the division is only performed for valid inputs and that the denominator is never zero.

Graphically, function division can produce complex behaviors, especially near points where the denominator approaches zero. The graph of $rac{f(x)}{g(x)}$ will show how the ratio of the functions behaves. Near values where $g(x)$ is close to zero, the quotient can become very large (positive or negative), leading to vertical asymptotes. The behavior of the quotient is also influenced by the signs and magnitudes of $f(x)$ and $g(x)$. Visualizing function division often requires careful analysis of the behavior of both functions and their ratio.

Evaluating Combined Functions

In this section, we will delve into the practical application of evaluating combined functions. By applying the operations discussed earlier, we will work through examples to illustrate how to find the values of these combined functions for specific inputs. This hands-on approach will solidify your understanding and enhance your ability to solve a variety of problems involving combined functions.

5. Addition of a Constant and a Function: $(90+f)(x)$

Adding a constant to a function involves adding the constant value to the function's expression. To find $(90+f)(x)$, you simply add the constant $90$ to the expression of the function $f(x)$. This operation shifts the graph of the function vertically by the value of the constant.

For example, if $f(x) = x^2 - 3x + 2$, then $(90+f)(x)$ is found by adding $90$ to the expression of $f(x)$: $(90 + f)(x) = 90 + (x^2 - 3x + 2)$. Combining like terms simplifies the expression to $x^2 - 3x + 92$. The resulting function, $x^2 - 3x + 92$, represents a vertical shift of the original function $f(x)$ by $90$ units upward.

The domain of the resulting function remains the same as the domain of the original function because adding a constant does not introduce any new restrictions on the input values. If $f(x)$ is defined for all real numbers, then $(90+f)(x)$ is also defined for all real numbers. This simplicity in domain considerations makes addition of a constant a straightforward operation.

Graphically, adding a constant to a function results in a vertical translation of the graph. The graph of $(90+f)(x)$ is the same as the graph of $f(x)$ but shifted upward by $90$ units. This means that each point on the graph of $f(x)$ is moved vertically upward by $90$ units to produce the graph of $(90+f)(x)$. This vertical shift preserves the shape and characteristics of the original function, only changing its vertical position.

6. Multiplication of a Constant and a Function: $(90f)(1)$

Multiplying a function by a constant involves multiplying the entire expression of the function by the constant value. To find $(90f)(1)$, you first multiply the expression of the function $f(x)$ by $90$, and then evaluate the resulting function at $x = 1$. This operation scales the function vertically by the constant factor.

For example, if $f(x) = x^3 - 2x^2 + x - 4$, then $90f(x)$ is found by multiplying the entire expression of $f(x)$ by $90$: $90f(x) = 90(x^3 - 2x^2 + x - 4)$. Distributing the $90$ across the terms inside the parentheses yields $90x^3 - 180x^2 + 90x - 360$. To evaluate $(90f)(1)$, we substitute $x = 1$ into this expression: $90(1)^3 - 180(1)^2 + 90(1) - 360$, which simplifies to $90 - 180 + 90 - 360$. The final result is $-360$. This value represents the scaled value of the function at $x = 1$.

The domain of the resulting function remains the same as the domain of the original function, as multiplying by a constant does not introduce any new restrictions on the input values. If $f(x)$ is defined for all real numbers, then $90f(x)$ is also defined for all real numbers. This consistency in domain makes multiplication by a constant a straightforward operation in terms of input values.

Graphically, multiplying a function by a constant results in a vertical stretch or compression of the graph, depending on the value of the constant. If the constant is greater than $1$, the graph is stretched vertically, making it taller. If the constant is between $0$ and $1$, the graph is compressed vertically, making it shorter. If the constant is negative, the graph is also reflected across the x-axis. In the case of $90f(x)$, the graph of $f(x)$ would be stretched vertically by a factor of $90$, making it significantly taller.

Conclusion

In conclusion, understanding and evaluating combined functions is a crucial skill in mathematics. Whether it's adding, subtracting, multiplying, or dividing functions, each operation has its unique properties and considerations. By mastering these operations, you can effectively manipulate functions, solve complex problems, and gain a deeper understanding of mathematical relationships. This guide has provided a comprehensive overview of function operations and evaluations, equipping you with the knowledge and tools needed to confidently tackle function-related challenges.