Rational Functions Exercises

Subsection 1.4 Rational Functions Exercises

Refer to $A(x)=\frac{4x+5}{2x-4}\text{.}$
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1. What is the domain of $A\text{?}$
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2. Graph $A(x)$ so that the important features are visible, and label them.
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3. Let $B(x)$ be the function whose graph is a shift of $A(x)$ down 3 units, and to the right 5 units. Write $B(x)$ as a transformation in terms of $A(x)\text{,}$ and then use that to get an algebraic formula for $B(x)\text{.}$
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Refer to $T(x)=\frac{(x+1)^2}{x+1}\text{.}$
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1. What is the domain of $T\text{?}$
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2. Graph $T(x)$ so that the important features are visible, and label them.
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3. In Desmos, use the settings to view a table of values for $T(x)\text{.}$ How is the domain reflected in the table?
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4. Is the graph what you expected? Explain.
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Refer to the function $G(x)=\frac{2x+3}{(x+4)(x-2)}\text{.}$
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1. Graph $G(x)$ so that the important features are visible, and label them.
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2. What is the domain of $G\text{?}$
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3. On which interval(s) is $G(x)$ decreasing?
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4. Let $F(x)=G(x)-3\text{.}$ Graph $F(x)$ and label its features.
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A toy rocket is launched vertically into the air from a height of $h$ meters and with an initial upward velocity of $v$ meters/second. The ball’s height above ground is given by the equation $H(t)=-4.9t^2 + vt + h\text{,}$ where $H$ is in meters and $t$ is in seconds. (This is the metric version of the gravity model.)
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1. Write an equation to model the height of a rocket launched from a height of 2 meters off the ground, with an initial upward velocity of 25 meters/second.
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2. How long is the ball in the air?
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3. What is the maximum height reached by the ball? When does it reach this height?
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4. When is the rocket at a height of 10 meters?
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5. When is the rocket at a height of greater than 20 meters?
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Let $a(x)=e^x\text{,}$ $b(x)=e^{-x+2}\text{,}$ $c(x)=\frac{1}{x^2}\text{.}$ Write a function equation for each function below, using the exponent rules to simplify the equation and write without negative exponents.
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1. $\displaystyle k(x)=(c \circ b)(x)$
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2. $\displaystyle f(x)=c(x) \cdot a(x)$
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3. $\displaystyle g(x)=(b(x))^{-2}$
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4. $\displaystyle h(x)=2 \left( \frac{a(x)}{b(x)} \right)$
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5. $\displaystyle j(x)=a(x)\cdot b(x)$
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Write as a single logarithm: $2\log_5{x}-5\log_5{y-5}+\frac{1}{3}\log_5{z}$
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Write as a sum and/or difference of logarithms: $\ln \left(\frac{x^4y^3}{z^2}\right)$
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Refer to the graph of $f(x)$ in Figure 1.4.12. Let $g(x)=-f(x-3)+1\text{.}$
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$A graph showing a piecewise function on a coordinate grid. The curve starts at $(-2,0)\text{,}$ rises smoothly to $(0, 4)\text{,}$ then remains constant at $y = 4$ until $x = 3\text{.}$ After that, the graph decreases linearly to $(5, 0)\text{.}$ The visible grid shows x-values from about -2 to 6 and y-values from about -2 to 6.$

Figure 1.4.12. Graph of a piecewise-defined function.
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1. Describe the transformations that would be required to transform $f(x)$ in to $g(x)\text{.}$
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2. On the same set of axes, sketch the graph of $g(x)\text{.}$
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3. What are the domain and range of $f(x)\text{?}$
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4. What are the domain and range of $g(x)\text{?}$
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