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Problem 10

# Let $$f$$ be the function defined by $$f(x)=2+2 \sqrt{5-x}$$. Find $f(-4), f(1), f\left(\frac{11}{4}\right)$$, and$$f(x+5)$.

Expert verified
In summary, we have found the following values for the function $$f(x) = 2+2\sqrt{5-x}$$: $$f(-4) = 8$$, $$f(1) = 6$$, $$f\left(\frac{11}{4}\right) = 5$$, and the new function $$f(x+5) = 2 + 2\sqrt{-x}$$.
See the step by step solution

## Step 1: Evaluate f(-4)

For this part, we need to find the value of the function f at point x = -4. To do that, we replace x with -4 in the equation and simplify it. $$f(-4) = 2 + 2\sqrt{5-(-4)}$$ $$f(-4) = 2 + 2\sqrt{5+4}$$ $$f(-4) = 2 + 2\sqrt{9}$$ $$f(-4) = 2 + 2(3)$$ $$f(-4) = 2 + 6$$ $$f(-4) = 8$$

## Step 2: Evaluate f(1)

For this part, we need to find the value of the function f at point x = 1. To do that, we replace x with 1 in the equation and simplify it. $$f(1) = 2 + 2\sqrt{5-1}$$ $$f(1) = 2 + 2\sqrt{4}$$ $$f(1) = 2 + 2(2)$$ $$f(1) = 2 + 4$$ $$f(1) = 6$$

## Step 3: Evaluate f(11/4)

For this part, we need to find the value of the function f at point x = 11/4. To do that, we replace x with 11/4 in the equation and simplify it. $$f\left(\frac{11}{4}\right) = 2 + 2\sqrt{5-\frac{11}{4}}$$ To simplify this, we can find a common denominator for 5 and 11/4: $$f\left(\frac{11}{4}\right) = 2 + 2\sqrt{\frac{20-11}{4}}$$ $$f\left(\frac{11}{4}\right) = 2 + 2\sqrt{\frac{9}{4}}$$ Now, take the square root of the fraction: $$f\left(\frac{11}{4}\right) = 2 + 2\left(\frac{3}{2}\right)$$ Finally, simplify: $$f\left(\frac{11}{4}\right) = 2 + 3$$ $$f\left(\frac{11}{4}\right) = 5$$

## Step 4: Evaluate f(x+5)

For this part, we need to create a new function by replacing x with x+5 in the original function and then simplify the equation. $$f(x+5) = 2 + 2\sqrt{5-(x+5)}$$ First, simplify the expression inside the square root: $$f(x+5) = 2 + 2\sqrt{5-x-5}$$ $$f(x+5) = 2 + 2\sqrt{-x}$$ So, the new function based on the given one is: $$f(x+5) = 2 + 2\sqrt{-x}$$

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