The range of f(x)=7*4^x is all positive real numbers is true.
What is the range of a function?
The range of a function is the set of its possible output values.
For the function f(x) on the domain of all real numbers,the range is the non-negative real numbers, which can be written as f(x)≥ 0.
Now the given function is-
f(x) = 7 x 4^x
Let y = f(x) = 7 x 4^x
=> y = 7 x 4^x
=> lny = ln {7 x 4^x}
Using logarithm equation,we get
=> ln y = ln7 + ln 4^x
=> ln y = ln7 + x ln4
=> y = e^(ln7 + x ln4)
=> y = e^ln7.e^(xln4)
Using exponential equation,we get
=> y = 7xln4
=> y = f(x) = 7.x.ln4
To find the value of function f(x)
7.x.ln7 = 0
Therefore, For all the set of values of x (x€R). We have
=> 7x ≥ 0
=> 7xln4 ≥ 0
Therefore interval notation will be (0,∞)
Set-builder notation :
{y|y>0}
By using all the values of x (x€R),the graph will be plot as-
Hence,the range of f(x)=7*4^x is all positive real numbers is true.
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