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Math  /  Algebra

QuestionLet f(x)=x+2x+7f−1(−3)=\begin{array}{l} f(x)=\frac{x+2}{x+7} \\ f^{-1}(-3)= \end{array}

Studdy Solution

STEP 1

1. We are given the function f(x)=x+2x+7 f(x) = \frac{x+2}{x+7} .
2. We need to find the inverse function f−1(x) f^{-1}(x) .
3. We need to evaluate f−1(−3) f^{-1}(-3) .

STEP 2

1. Set up the equation for the inverse function.
2. Solve for x x in terms of y y .
3. Express the inverse function f−1(x) f^{-1}(x) .
4. Substitute −3-3 into the inverse function.
5. Simplify to find the value of f−1(−3) f^{-1}(-3) .

STEP 3

Set up the equation for the inverse function by letting y=f(x) y = f(x) :
y=x+2x+7 y = \frac{x+2}{x+7}

STEP 4

Solve for x x in terms of y y . Start by cross-multiplying:
y(x+7)=x+2 y(x + 7) = x + 2

STEP 5

Distribute y y on the left side:
yx+7y=x+2 yx + 7y = x + 2

STEP 6

Rearrange terms to isolate x x :
yx−x=2−7y yx - x = 2 - 7y
Factor out x x from the left side:
x(y−1)=2−7y x(y - 1) = 2 - 7y

STEP 7

Solve for x x :
x=2−7yy−1 x = \frac{2 - 7y}{y - 1}

STEP 8

Express the inverse function f−1(x) f^{-1}(x) :
f−1(x)=2−7xx−1 f^{-1}(x) = \frac{2 - 7x}{x - 1}

STEP 9

Substitute −3-3 into the inverse function:
f−1(−3)=2−7(−3)−3−1 f^{-1}(-3) = \frac{2 - 7(-3)}{-3 - 1}

STEP 10

Simplify the expression:
Calculate the numerator:
2−7(−3)=2+21=23 2 - 7(-3) = 2 + 21 = 23
Calculate the denominator:
−3−1=−4 -3 - 1 = -4
Thus,
f−1(−3)=23−4=−234 f^{-1}(-3) = \frac{23}{-4} = -\frac{23}{4}
The value of f−1(−3) f^{-1}(-3) is:
−234 \boxed{-\frac{23}{4}}

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