Problem 44 - Calculus with Concepts in Calculus

Question

In each of the following, determine whether \(f\) and \(g\) are the same. a. \(f(x)=1-x^{2} ; g(x)=1-x^{2}\) for \(-1

Step-by-Step Solution

Answer

Parts a, b, and c have the same functions; parts d, e, and f do not share the same functions in all conditions.

Step 1: Analyze functions for part a

Given functions: \(f(x)=1-x^{2}\) and \(g(x)=1-x^{2}\) for \(-1

Step 2: Analyze functions for part b

Given functions: \(f(x)=\sqrt{x}\) for \(x \geq 0\) and \(g(x)=\sqrt{x}\). Both functions are the same; they both represent the square root of \(x\) for non-negative values.

Step 3: Analyze functions for part c

Given functions: \(f(x)=\sqrt{x^{2}}\) and \(g(x)=|x|\). The function \(f(x)\) simplifies to \(|x|\) for all real \(x\), hence \(f(x)\) and \(g(x)\) are the same.

Step 4: Analyze functions for part d

Given functions: \(f(x)=\frac{x^{3}-4x}{x^{3}-4x}\) and \(g(x)=1\). The function \(f(x)\) equals 1 wherever defined except at points where \(x^{3}-4x=0\). The equation \(x(x^2-4)=0\) has solutions \(x=0, 2, -2\). Thus, \(f(x) eq g(x)\) at \(x=0, 2, -2\). Therefore, they are not the same.

Step 5: Analyze functions for part e

Given functions: \(f(x)=\frac{x-1}{x^{2}-1}\) and \(g(x)=\frac{1}{x+1}\). Simplify \(f(x)\): \(x^2-1 = (x-1)(x+1)\), so \(f(x) = \frac{x-1}{(x-1)(x+1)} = \frac{1}{x+1}\) when \(xeq 1\). Given restrictions match, therefore \(f(x)\) and \(g(x)\) are the same except possibly at \(x=1\). Their domains differ only at \(x=1\).

Step 6: Analyze functions for part f

Given functions: \( f(x) = \frac{x^{2} - 5x + 6}{x+2} \) and \( g(x) = x-3 \), for \( x eq -2 \). Simplify \( f(x) \): \( x^{2}-5x+6 = (x-3)(x-2) \), so \( f(x) = \frac{(x-3)(x-2)}{x+2} \). However, \(f(x)\) cannot be simplified to \(g(x)\), thus \( f(x) \) and \( g(x) \) are not the same.

Key Concepts

Domain Analysis

Understanding the domain of a function is crucial in determining whether two functions are equivalently defined. The domain of a function refers to all the possible input values (commonly represented as 'x' values) for which the function is defined and provides a valid output. For instance, in problem part a, both functions have the same formula,

• \(f(x) = g(x) = 1-x^{2}\)

Valid for the domain

• \(-1 < x < 1\).

It is the definition inside the same domain that makes these functions equivalent.
Similarly, knowing the domain helps to identify when functions diverge in behaviors, like in part d, where the domain excludes points that result in an undefined form \(\left(\frac{0}{0}\right)\). Thus, it's essential to analyze the domains to determine when two functions might be the same or different based on where they "function".

Function Simplification

When comparing functions, simplification helps us see the equivalence or distinction between them. Simplifying involves reducing the function to its simplest form by eliminating unnecessary parts, like common factors in rational expressions.
For instance, in part e, the function

• \(f(x)=\frac{x-1}{x^{2}-1}\)

can be simplified. The denominator \(x^{2} - 1\)

• Factorizes into \((x-1)(x+1)\),

leading to a simplified form

• \(f(x)=\frac{1}{x+1}\)

for \(x eq 1\). Such simplification reveals that \(f(x)\) is essentially equivalent to \(g(x)\), provided their domains align or differ only by valid points.
Simplification plays a critical role, especially for rational functions, to eliminate indeterminacies while discernibly presenting essential characteristics of the functions involved.

Equivalence of Functions

Determining equivalence means checking if two functions produce identical outputs for all values within their domains. Two functions might appear different initially but could return to identical outputs when simplified or carefully analyzed against their domains.
For example, in part c, the expression

• \(f(x)=\sqrt{x^{2}}\)

is assessed against

• \(g(x)=|x|\).

In this instance, they are equivalent because \( \sqrt{x^2} \) is the same as \(|x|\) for all real numbers. By understanding this equivalence, we see how transformations or specific operations, such as the absolute value, maintain functional behavior despite apparent differences.
This core focus on equivalence ensures we understand when seemingly different expressions represent the same mathematical reality.

Rational Functions

Rational functions are those that are expressed as the ratio of two polynomials. These functions are particularly sensitive to their domains since divisions by zero are undefined.
In problems like step d,

• The function \(f(x) = \frac{x^{3}-4x}{x^{3}-4x}\)

is comparable to

• \(g(x)=1\).

When simplifying, we understand that \(f(x)=1\) for all points except where the denominator equals zero, namely \(x = 0, 2, -2\). This identifies domain restrictions that make \(f(x)eq g(x)\) at certain points, hence not equivalent overall.
Rational functions invite intricacies of both algebraic manipulation and a clear understanding of where each functional form is defined and valid.

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Problem 44

Other exercises in this chapter

Problem 44

Evaluate the expression. $$ |-\sqrt{2}|^{2} $$

View solution

Problem 44

Find an equation of the line that is perpendicular to the given line \(l\) and passes through the given point \(P\). \(l: y=-\frac{1}{3} x-2 ; P=(0,0)\)

View solution

Problem 44

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ |x|=|y| $$

View solution

Problem 45

Approximate all zeros of the function to the nearest hundredth. $$ f(x)=-4.9 x^{2}+5.1 x+1.2 $$

View solution

Key Concepts

Domain Analysis

Understanding the domain of a function is crucial in determining whether two functions are equivalently defined. The domain of a function refers to all the possible input values (commonly represented as 'x' values) for which the function is defined and provides a valid output. For instance, in problem part a, both functions have the same formula,

• \(f(x) = g(x) = 1-x^{2}\)

Valid for the domain

• \(-1 < x < 1\).

It is the definition inside the same domain that makes these functions equivalent.
Similarly, knowing the domain helps to identify when functions diverge in behaviors, like in part d, where the domain excludes points that result in an undefined form \(\left(\frac{0}{0}\right)\). Thus, it's essential to analyze the domains to determine when two functions might be the same or different based on where they "function".

Function Simplification

When comparing functions, simplification helps us see the equivalence or distinction between them. Simplifying involves reducing the function to its simplest form by eliminating unnecessary parts, like common factors in rational expressions.
For instance, in part e, the function

• \(f(x)=\frac{x-1}{x^{2}-1}\)

can be simplified. The denominator \(x^{2} - 1\)

• Factorizes into \((x-1)(x+1)\),

leading to a simplified form

• \(f(x)=\frac{1}{x+1}\)

for \(x eq 1\). Such simplification reveals that \(f(x)\) is essentially equivalent to \(g(x)\), provided their domains align or differ only by valid points.
Simplification plays a critical role, especially for rational functions, to eliminate indeterminacies while discernibly presenting essential characteristics of the functions involved.

Equivalence of Functions

Determining equivalence means checking if two functions produce identical outputs for all values within their domains. Two functions might appear different initially but could return to identical outputs when simplified or carefully analyzed against their domains.
For example, in part c, the expression

• \(f(x)=\sqrt{x^{2}}\)

is assessed against

• \(g(x)=|x|\).

In this instance, they are equivalent because \( \sqrt{x^2} \) is the same as \(|x|\) for all real numbers. By understanding this equivalence, we see how transformations or specific operations, such as the absolute value, maintain functional behavior despite apparent differences.
This core focus on equivalence ensures we understand when seemingly different expressions represent the same mathematical reality.

Rational Functions

Rational functions are those that are expressed as the ratio of two polynomials. These functions are particularly sensitive to their domains since divisions by zero are undefined.
In problems like step d,

• The function \(f(x) = \frac{x^{3}-4x}{x^{3}-4x}\)

is comparable to

• \(g(x)=1\).

When simplifying, we understand that \(f(x)=1\) for all points except where the denominator equals zero, namely \(x = 0, 2, -2\). This identifies domain restrictions that make \(f(x)eq g(x)\) at certain points, hence not equivalent overall.
Rational functions invite intricacies of both algebraic manipulation and a clear understanding of where each functional form is defined and valid.