Given functions f(x) = 2x + 7 and g(x) = 3/(x−2), find f⁻¹(x), gf(x), and gf(−2).

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Here are the solutions for the given functions: Given functions: f(x) = 2x + 7; g(x) = (3)/(x−2) a) Find f⁻¹(x) Step 1: Set y = f(x). y = 2x + 7 Step 2: Swap x and y. x = 2y + 7 Step 3: Solve for y. Subtract 7 from both sides: x − 7 = 2y Divide by 2: y = (x − 7)/(2) Step 4: Replace y with f⁻¹(x). f⁻¹(x) = (x − 7)/(2); The inverse function is f⁻¹(x) = (x − 7)/(2). b) Find gf(x) Step 1: Substitute f(x) into g(x). The definition of gf(x) is g(f(x)). Replace x in g(x) with f(x). gf(x) = g(2x + 7) Step 2: Apply the definition of g(x). gf(x) = (3)/((2x + 7) − 2) Step 3: Simplify the expression. gf(x) = (3)/(2x + 5); The composite function is gf(x) = (3)/(2x + 5). c) Find gf(−2) Step 1: Use the expression for gf(x) found in part b. gf(x) = (3)/(2x + 5); Step 2: Substitute x = −2 into the expression. gf(−2) = (3)/(2(−2) + 5) Step 3: Simplify the denominator. gf(−2) = (3)/(−4 + 5); gf(−2) = 3/1; gf(−2) = 3 The value of gf(−2) is 3.

Given functions f(x) = 2x + 7 and g(x) = 3/(x-2), find f⁻¹(x), gf(x), and gf(-2).

Answers

Step-by-step solution

Part a· 4 steps

Set $y = f(x)$ .

Swap $x$ and $y$ .

Solve for $y$ .

Replace $y$ with $f^{-1}(x)$ .

Part b· 3 steps
$gf(x) = \frac{3}{2x + 5}$

Substitute $f(x)$ into $g(x)$ .

Apply the definition of $g(x)$ .

Simplify the expression.

Part c· 3 steps3

Use the expression for $gf(x)$ found in part b.

Substitute $x = -2$ into the expression.

Simplify the denominator.

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More Mathematics Questions

Given functions f(x) = 2x + 7 and g(x) = 3/(x-2), find f⁻¹(x), gf(x), and gf(-2).

Answers

Step-by-step solution

Part a· 4 steps

Set y=f(x)y = f(x)y=f(x).

Swap xxx and yyy.

Solve for yyy.

Replace yyy with f−1(x)f^{-1}(x)f−1(x).

Part b· 3 stepsgf(x)=32x+5gf(x) = \frac{3}{2x + 5}gf(x)=2x+53​

Substitute f(x)f(x)f(x) into g(x)g(x)g(x).

Apply the definition of g(x)g(x)g(x).