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Question
y=2x+3
Function
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Find the slope
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Find the inverse
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Evaluate the derivative
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Find the domain
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\text{Find the }x\text{-intercept/zero}
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Find the y-intercept
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Find the range
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Find the vertical asymptotes
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Find the horizontal asymptotes
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Determine if even, odd or neither
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Find the stationary points
More methods
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m=2
Evaluate
y=2x+3
Solution
m=2
f^{-1}\left(x\right) = \frac{x-3}{2}
Evaluate
y=2x+3
\text{Interchange }x\text{ and }y
x=2y+3
Swap the sides of the equation
2y+3=x
Move the constant to the right-hand side and change its sign
2y=x-3
Divide both sides
\frac{2y}{2}=\frac{x-3}{2}
Divide the numbers
y=\frac{x-3}{2}
Solution
f^{-1}\left(x\right) = \frac{x-3}{2}
y^{\prime}=2
Evaluate
y=2x+3
Take the derivative of both sides
y^{\prime}=\frac{d}{dx}\left(2x+3\right)
\text{Use differentiation rule }\frac{d}{dx}\left(f\left(x\right)\pm g\left(x\right)\right)=\frac{d}{dx}\left(f\left(x\right)\right)\pm \frac{d}{dx}(g(x))
y^{\prime}=\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(3\right)
Calculate
More Steps
Calculate
\frac{d}{dx}\left(2x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
2\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2\times 1
Any expression multiplied by 1 remains the same
2
y^{\prime}=2+\frac{d}{dx}\left(3\right)
\text{Use }\frac{d}{dx}(c)=0\text{ to find derivative}
y^{\prime}=2+0
Solution
y^{\prime}=2
x \in \mathbb{R}
Evaluate
y=2x+3
Separate the function into parts to determine the domain of each part
2x+3
Solution
x \in \mathbb{R}
x=-\frac{3}{2}
Evaluate
y=2x+3
\text{To find the }x\text{-intercept,set }y\text{=0}
0=2x+3
Swap the sides of the equation
2x+3=0
Move the constant to the right-hand side and change its sign
2x=0-3
Removing 0 doesn't change the value,so remove it from the expression
2x=-3
Divide both sides
\frac{2x}{2}=\frac{-3}{2}
Divide the numbers
x=\frac{-3}{2}
Solution
x=-\frac{3}{2}
y=3
Evaluate
y=2x+3
\text{To find the }y\text{-intercept,set }x\text{=0}
y=2\times 0+3
Any expression multiplied by 0 equals 0
y=0+3
Solution
y=3
y \in \mathbb{R}
Evaluate
y=2x+3
Solution
y \in \mathbb{R}
\textrm{No vertical asymptotes}
Evaluate
y=2x+3
The domain of a polynomial function is the set of all real numbers
y=2x+3,x \in \mathbb{R}
Solution
\textrm{No vertical asymptotes}
\textrm{No horizontal asymptotes}
Evaluate
y=2x+3
\text{To determine the horizontal asymptote,evaluate the limits }\lim _{x\rightarrow +\infty}\left(y\right)\text{ and }\lim _{x\rightarrow -\infty}\left(y\right)
\begin{align}&\lim _{x\rightarrow +\infty}\left(2x+3\right)\\&\lim _{x\rightarrow -\infty}\left(2x+3\right)\end{align}
Calculate
More Steps
Evaluate
\lim _{x\rightarrow +\infty}\left(2x+3\right)
Rewrite the expression
\lim _{x\rightarrow +\infty}\left(2x\right)+\lim _{x\rightarrow +\infty}\left(3\right)
Calculate
More Steps
Evaluate
\lim _{x\rightarrow +\infty}\left(2x\right)
Rewrite the expression
2\times \lim _{x\rightarrow +\infty}\left(x\right)
Calculate
2\left(+\infty\right)
Calculate
+\infty
\left(+\infty\right)+\lim _{x\rightarrow +\infty}\left(3\right)
Calculate
\left(+\infty\right)+3
Calculate
+\infty
\begin{align}&+\infty\\&\lim _{x\rightarrow -\infty}\left(2x+3\right)\end{align}
Calculate
More Steps
Evaluate
\lim _{x\rightarrow -\infty}\left(2x+3\right)
Rewrite the expression
\lim _{x\rightarrow -\infty}\left(2x\right)+\lim _{x\rightarrow -\infty}\left(3\right)
Calculate
More Steps
Evaluate
\lim _{x\rightarrow -\infty}\left(2x\right)
Rewrite the expression
2\times \lim _{x\rightarrow -\infty}\left(x\right)
Calculate
2\left(-\infty\right)
Calculate
-\infty
\left(-\infty\right)+\lim _{x\rightarrow -\infty}\left(3\right)
Calculate
\left(-\infty\right)+3
Calculate
-\infty
\begin{align}&+\infty\\&-\infty\end{align}
Solution
\textrm{No horizontal asymptotes}
\textrm{Neither even nor odd}
Evaluate
y=2x+3
To determine if the function is even or odd,use the appropriate notation
f\left(x\right)=2x+3
\text{Substitute }-x\text{ for }x
f\left(-x\right)=2\left(-x\right)+3
Simplify
f\left(-x\right)=-2x+3
Solution
\textrm{Neither even nor odd}
\text{No stationary points}
Evaluate
y=2x+3
Solution
\text{No stationary points}
Show Solutions
Testing for symmetry
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Testing for symmetry about the origin
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Testing for symmetry about the x-axis
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Testing for symmetry about the y-axis
\textrm{Not symmetry with respect to the origin}
Evaluate
y=2x+3
\text{To test if the graph of }y=2x+3\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
-y=2\left(-x\right)+3
Simplify
-y=-2x+3
Change the signs both sides
y=2x-3
Solution
\textrm{Not symmetry with respect to the origin}
\textrm{Not symmetry with respect to the x-axis}
Evaluate
y=2x+3
\text{To test if the graph of }y=2x+3\text{ is symmetry with respect to the x-axis,substitute -y for y}
-y=2x+3
Change the signs both sides
y=-2x-3
Solution
\textrm{Not symmetry with respect to the x-axis}
\textrm{Not symmetry with respect to the y-axis}
Evaluate
y=2x+3
\text{To test if the graph of }y=2x+3\text{ is symmetry with respect to the y-axis,substitute -x for x}
y=2\left(-x\right)+3
Simplify
y=-2x+3
Solution
\textrm{Not symmetry with respect to the y-axis}
Show Solutions
Solve the equation
x=\frac{y-3}{2}
Evaluate
y=2x+3
Swap the sides of the equation
2x+3=y
Move the constant to the right-hand side and change its sign
2x=y-3
Divide both sides
\frac{2x}{2}=\frac{y-3}{2}
Solution
x=\frac{y-3}{2}
Show Solutions
Rewrite the equation
r=\frac{3}{\sin\left(\theta \right)-2\cos\left(\theta \right)}
Evaluate
y=2x+3
Move the expression to the left side
y-2x=3
\text{To convert the equation to polar coordinates,substitute }r\cos\left(\theta \right)\text{ for }x\text{ and }r\sin\left(\theta \right)\text{ for }y
\sin\left(\theta \right)\times r-2\cos\left(\theta \right)\times r=3
Factor the expression
\left(\sin\left(\theta \right)-2\cos\left(\theta \right)\right)r=3
Solution
r=\frac{3}{\sin\left(\theta \right)-2\cos\left(\theta \right)}
Show Solutions
Graph
