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Math  /  Algebra

QuestionExamples 2 and 3 Find the inverse of each function. Then graph the function and its inverse. If necessary, restrict the domain of the inverse so that it is a function.
5. f(x)=x+2f(x)=x+2
6. g(x)=5xg(x)=5 x
7. f(x)=−2x+1f(x)=-2 x+1
8. h(x)=x−43h(x)=\frac{x-4}{3}
9. f(x)=−53x−8f(x)=-\frac{5}{3} x-8
10. g(x)=x+4g(x)=x+4
11. f(x)=4xf(x)=4 x
12. f(x)=−8x+9f(x)=-8 x+9
13. f(x)=5x2f(x)=5 x^{2}
14. h(x)=x2+4h(x)=x^{2}+4

Studdy Solution

STEP 1

1. We need to find the inverse of each given function.
2. We need to graph both the function and its inverse.
3. If the inverse is not a function over the entire domain, we may need to restrict the domain.

STEP 2

1. Find the inverse of each function.
2. Determine if the inverse is a function, and if necessary, restrict the domain.
3. Graph the function and its inverse.

STEP 3

For each function, find the inverse by solving for x x in terms of y y .
- For f(x)=x+2 f(x) = x + 2 : y=x+2 y = x + 2 Solve for x x : x=y−2 x = y - 2 So, the inverse is f−1(x)=x−2 f^{-1}(x) = x - 2 .
- For g(x)=5x g(x) = 5x : y=5x y = 5x Solve for x x : x=y5 x = \frac{y}{5} So, the inverse is g−1(x)=x5 g^{-1}(x) = \frac{x}{5} .
- For f(x)=−2x+1 f(x) = -2x + 1 : y=−2x+1 y = -2x + 1 Solve for x x : x=1−y2 x = \frac{1 - y}{2} So, the inverse is f−1(x)=1−x2 f^{-1}(x) = \frac{1 - x}{2} .
- For h(x)=x−43 h(x) = \frac{x-4}{3} : y=x−43 y = \frac{x-4}{3} Solve for x x : x=3y+4 x = 3y + 4 So, the inverse is h−1(x)=3x+4 h^{-1}(x) = 3x + 4 .
- For f(x)=−53x−8 f(x) = -\frac{5}{3}x - 8 : y=−53x−8 y = -\frac{5}{3}x - 8 Solve for x x : x=−35(y+8) x = -\frac{3}{5}(y + 8) So, the inverse is f−1(x)=−35(x+8) f^{-1}(x) = -\frac{3}{5}(x + 8) .
- For g(x)=x+4 g(x) = x + 4 : y=x+4 y = x + 4 Solve for x x : x=y−4 x = y - 4 So, the inverse is g−1(x)=x−4 g^{-1}(x) = x - 4 .
- For f(x)=4x f(x) = 4x : y=4x y = 4x Solve for x x : x=y4 x = \frac{y}{4} So, the inverse is f−1(x)=x4 f^{-1}(x) = \frac{x}{4} .
- For f(x)=−8x+9 f(x) = -8x + 9 : y=−8x+9 y = -8x + 9 Solve for x x : x=9−y8 x = \frac{9 - y}{8} So, the inverse is f−1(x)=9−x8 f^{-1}(x) = \frac{9 - x}{8} .
- For f(x)=5x2 f(x) = 5x^2 : y=5x2 y = 5x^2 Solve for x x : x=±y5 x = \pm \sqrt{\frac{y}{5}} The inverse is not a function unless we restrict the domain. Choose x≥0 x \geq 0 or x≤0 x \leq 0 .
- For h(x)=x2+4 h(x) = x^2 + 4 : y=x2+4 y = x^2 + 4 Solve for x x : x=±y−4 x = \pm \sqrt{y - 4} The inverse is not a function unless we restrict the domain. Choose x≥0 x \geq 0 or x≤0 x \leq 0 .

STEP 4

Determine if the inverse is a function and restrict the domain if necessary.
- For f(x)=5x2 f(x) = 5x^2 and h(x)=x2+4 h(x) = x^2 + 4 , restrict the domain to x≥0 x \geq 0 or x≤0 x \leq 0 to make the inverse a function.

STEP 5

Graph each function and its inverse. The graph of a function and its inverse are reflections over the line y=x y = x .
- Plot the original function. - Plot the inverse function. - Draw the line y=x y = x to show reflection.

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