Questions: Determine whether the following statements are true and give an explanation or counterexample. a. All polynomials are rational functions but not all rational functions are polynomials. b. If f is a linear polynomial, then f o f is a quadratic polynomial. c. If f and g are polynomials, then the degrees of f ∘ g and g ∘ f are equal. d. To graph g(x)=f(x+2), shift the graph of f two units to the right. b. Choose the correct answer below. A. The statement is true. Let f(x)=ax+b be a linear function where a and b are constants. Then f ∘ f=(ax+b)(ax+b)=a^2 x^2+2abx+b^2. B. The statement is true. The degree of the composition function f ∘ g is the sum or the degrees of the functions f and g. If both functions are linear, the sum of their degrees is 2. C. The statement is false. Let f(x)=x+2. Then f ∘ f=f(f(x))=(x+2)+2=x+4. This is a linear function.

Determine whether the following statements are true and give an explanation or counterexample.
a. All polynomials are rational functions but not all rational functions are polynomials.
b. If f is a linear polynomial, then f o f is a quadratic polynomial.
c. If f and g are polynomials, then the degrees of f ∘ g and g ∘ f are equal.
d. To graph g(x)=f(x+2), shift the graph of f two units to the right.
b. Choose the correct answer below.
A. The statement is true. Let f(x)=ax+b be a linear function where a and b are constants. Then f ∘ f=(ax+b)(ax+b)=a^2 x^2+2abx+b^2.
B. The statement is true. The degree of the composition function f ∘ g is the sum or the degrees of the functions f and g. If both functions are linear, the sum of their degrees is 2.
C. The statement is false. Let f(x)=x+2. Then f ∘ f=f(f(x))=(x+2)+2=x+4. This is a linear function.

Transcript text: Determine whether the following statements are true and give an explanation or counterexample. a. All polynomials are rational functions but not all rational functions are polynomials. b. If $f$ is a linear polynomial, then $f$ o $f$ is a quadratic polynomial. c. If $f$ and $g$ are polynomials, then the degrees of $f \circ g$ and $g \circ f$ are equal. d. To graph $g(x)=f(x+2)$, shift the graph of $f$ two units to the right. b. Choose the correct answer below. A. The statement is true. Let $f(x)=a x+b$ be a linear function where $a$ and $b$ are constants. Then $f \circ f=(a x+b)(a x+b)=a^{2} x^{2}+2 a b x+b^{2}$. B. The statement is true. The degree of the composition function $f \circ g$ is the sum or the degrees of the functions $f$ and $g$. If both functions are linear, the sum of their degrees is 2. C. The statement is false. Let $f(x)=x+2$. Then $f \circ f=f(f(x))=(x+2)+2=x+4$. This is a linear function.

Solution

Solution Steps

To determine the truth of each statement, we need to analyze them individually:

a. A polynomial is a specific type of rational function where the denominator is 1. However, rational functions can have any polynomial in the denominator, so not all rational functions are polynomials.

b. If $ f(x) = ax + b $, then $ f(f(x)) = a(ax + b) + b = a^2x + ab + b $, which is indeed a quadratic polynomial.

c. The degree of $ f \circ g $ is the product of the degrees of $ f $ and $ g $, not necessarily equal to $ g \circ f $.

d. To graph $ g(x) = f(x+2) $, shift the graph of $ f $ two units to the left, not right.

Step 1: Analyze Statement a

A polynomial is a rational function where the denominator is 1.
A rational function can have any polynomial in the denominator, so not all rational functions are polynomials.

Step 2: Analyze Statement b

Let $ f(x) = ax + b $.
Then $ f(f(x)) = a(ax + b) + b = a^2x + ab + b $.
This is a quadratic polynomial.

Step 3: Analyze Statement c

The degree of $ f \circ g $ is the product of the degrees of $ f $ and $ g $.
The degree of $ g \circ f $ is also the product of the degrees of $ g $ and $ f $.
These degrees are equal only if the degrees of $ f $ and $ g $ are the same.

Step 4: Analyze Statement d

To graph $ g(x) = f(x+2) $, shift the graph of $ f $ two units to the left, not right.

Final Answer

Statement a: True, all polynomials are rational functions but not all rational functions are polynomials.
Statement b: True, $ f \circ f $ is a quadratic polynomial.
Statement c: False, the degrees of $ f \circ g $ and $ g \circ f $ are not necessarily equal.
Statement d: False, $ g(x) = f(x+2) $ shifts the graph left.

\[ \boxed{\text{a: True, b: True, c: False, d: False}} \]

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