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Exponential Functions

Exponential Functions 1.8 - Solution

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We want to identify the value of the base of the exponential function from the given graph.

Let's consider the given equation for this graph. f(x)=bx\begin{gathered} f(x)=b^x \end{gathered} Let's first substitute the given point which has 11 as its x-x\text{-}coordinate into the above formula. This will allow us to find the value of b.b. Let's substitute (1,5)(1,5) into the above formula.
f(x)=bxf(x)=b^x
5=b1{\color{#009600}{5}}=b^{\color{#0000FF}{1}}
Solve for bb
5=b5=b
b=5b=5
Now we can write our equation. f(x)=bxf(x)=5x\begin{gathered} f(x)={\color{#FF0000}{b}}^x \quad \Rightarrow \quad f(x)={\color{#FF0000}{5}}^x \end{gathered} Next, we will substitute other given points, (0,1)(0,1) and (-1,15),\left(\text{-} 1, \frac{1}{5}\right), into the above equation to check it. Let's start from (0,1).(0,1).
f(x)=5xf(x)=5^x
1=50{\color{#009600}{1}}=5^{\color{#0000FF}{0}}
1=11=1
The second point satisfies the equation. Let's try the last one, (-1,15).\left(\text{-} 1, \frac{1}{5}\right).
f(x)=5xf(x)=5^x
15=5-1{\color{#009600}{\dfrac{1}{5}}}=5^{\color{#0000FF}{\text{-} 1}}
15=151\dfrac{1}{5}=\dfrac{1}{5^1}
15=15\dfrac{1}{5}=\dfrac{1}{5}
All the given points satisfy the equation. Now we can be sure that b=5.b=5.