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QuestionFind f(x)=x∣x∣f(x)=\frac{x}{|x|} for: a. f(6)f(6), b. f(−6)f(-6), c. f(r2)f\left(r^{2}\right).

Studdy Solution

STEP 1

Assumptions1. The function is defined as f(x)=x∣x∣f(x)=\frac{x}{|x|} . For part a, we are asked to find f(6)f(6)3. For part b, we are asked to find f(−6)f(-6)4. For part c, we are asked to find f(r)f\left(r^{}\right)5. We assume that rr is a real number

STEP 2

First, let's solve for f(6)f(6). We can do this by substituting x=6x=6 into the function.
f(6)=6∣6∣f(6)=\frac{6}{|6|}

STEP 3

Now, calculate the absolute value of6.
∣6∣=6|6| =6So, the function becomesf(6)=66f(6)=\frac{6}{6}

STEP 4

Calculate the value of f(6)f(6).
f(6)=66=1f(6)=\frac{6}{6}=1

STEP 5

Now, let's solve for f(−)f(-). We can do this by substituting x=−x=- into the function.
f(−)=−∣−∣f(-)=\frac{-}{|-|}

STEP 6

Now, calculate the absolute value of -6.
∣−6∣=6|-6| =6So, the function becomesf(−6)=−66f(-6)=\frac{-6}{6}

STEP 7

Calculate the value of f(−6)f(-6).
f(−6)=−66=−1f(-6)=\frac{-6}{6}=-1

STEP 8

Now, let's solve for f(r2)f\left(r^{2}\right). We can do this by substituting x=r2x=r^{2} into the function.
f(r2)=r2∣r2∣f\left(r^{2}\right)=\frac{r^{2}}{|r^{2}|}

STEP 9

Now, calculate the absolute value of r2r^{2}.
Since r2r^{2} is always non-negative for any real number rr, we have∣r2∣=r2|r^{2}| = r^{2}So, the function becomesf(r2)=r2r2f\left(r^{2}\right)=\frac{r^{2}}{r^{2}}

STEP 10

Calculate the value of f(r2)f\left(r^{2}\right).
f(r2)=r2r2=f\left(r^{2}\right)=\frac{r^{2}}{r^{2}}=So, the solutions area. f(6)=f(6)= b. f(−6)=−f(-6)=- c. f(r2)=f\left(r^{2}\right)=

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