Answer :

zayaanedu

Answer: A function that transforms the function f(x) = 4 √x + 1 such that f(x) is horizontally stretched by a factor of 5, reflected across the x-axis, then translated down 5 can be represented by the function:

g(x) = -(5(4 √(x/5)) + 1) - 5

Here's how each transformation is applied:

Horizontal stretching by a factor of 5: To stretch a function horizontally by a factor of 5, we divide the input variable x by 5 before applying the function. In this case, we divide x by 5 in the argument of the square root function: 4 √(x/5)

Reflection across the x-axis: To reflect a function across the x-axis, we multiply the output by -1. In this case, we multiply the output of the square root function by -1: -(4 √(x/5))

Translation down 5: To translate a function down 5 units, we subtract 5 from the output. In this case, we subtract 5 from the output of the square root function: -(4 √(x/5)) - 5

So, the final function g(x) = -(5(4 √(x/5)) + 1) - 5 that transform the function f(x) = 4 √x + 1 horizontally stretched by a factor of 5, reflected across the x-axis, then translated down 5.

Step-by-step explanation:

semsee45

Answer:

[tex]g(x)=-4\sqrt{\dfrac{x}{5}}-6[/tex]

Step-by-step explanation:

Given function:

[tex]f(x)=4\sqrt{x}+1[/tex]

1.  Horizontal stretch

[tex]f\left(\dfrac{1}{a}x\right) \implies f(x) \: \textsf{stretched parallel to the $x$-axis (horizontally) by a factor of $a$}.[/tex]

Therefore, if f(x) is horizontally stretched by a factor of 5:

[tex]\implies f\left(\dfrac{1}{5}x\right)=4\sqrt{\dfrac{x}{5}}+1[/tex]

2.  Reflection across the x-axis

[tex]-f(x)\implies f(x) \: \textsf{reflected in the $x$-axis}.[/tex]

Therefore, if f(x/5) is reflected in the x-axis:

[tex]\begin{aligned}\implies -f\left(\dfrac{1}{5}x\right)&=-\left(4\sqrt{\dfrac{x}{5}}+1\right)\\\\&=-4\sqrt{\dfrac{x}{5}}-1 \end{aligned}[/tex]

3.  Translation

[tex]f(x)-a \implies f(x) \: \textsf{translated $a$ units down}[/tex]

Therefore, if -f(x/5) is translated 5 units down:

[tex]\begin{aligned}\implies -f\left(\dfrac{1}{5}x\right)-5&=-4\sqrt{\dfrac{x}{5}}-1 -5\\\\&=-4\sqrt{\dfrac{x}{5}}-6\end{aligned}[/tex]

Therefore:

[tex]g(x)=-4\sqrt{\dfrac{x}{5}}-6[/tex]

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