Express f(x) in the form a + b/(x+1), where a and b are constants. Hence, give a sequence of three…

Question

Express f(x) in the form a + b/(x+1), where a and b are constants. Hence, give a sequence of three transformations which take the graph of y = 1/x onto the graph of y = f(x). State the range of f. Form the composite function f(f(x)). Hence, or otherwise, obtain an expression for f⁻¹(x).

ScanSolve AI Tutor · Accepted Answer

To express f(x) = (3x+2)/(x+1) in the form a + (b)/(x+1), we can use algebraic manipulation. Step 1: Rewrite the numerator in terms of the denominator. We want to create a term of (x+1) in the numerator. 3x+2 = 3(x+1) − 3 + 2; 3x+2 = 3(x+1) − 1 Step 2: Substitute this back into the expression for f(x). f(x) = (3(x+1) − 1)/(x+1) Step 3: Separate the terms. f(x) = (3(x+1))/(x+1) − (1)/(x+1); f(x) = 3 − (1)/(x+1) Step 4: Compare with the given form a + (b)/(x+1). By comparing f(x) = 3 − (1)/(x+1) with f(x) = a + (b)/(x+1), we find the values of a and b. a = 3; b = −1 The expression for f(x) in the required form is: f(x) = 3 − (1)/(x+1); where a=3 and b=−1.

Express f(x) in the form a + b/(x+1), where a and b are constants. Hence, give a sequence of three transformations which take the graph of y = 1/x onto the graph of y = f(x). State the range of f. Form the composite function f(f(x)). Hence, or otherwise, obtain an expression for f⁻¹(x).

Answer

Step-by-step solution

Rewrite the numerator in terms of the denominator.

Substitute this back into the expression for $f(x)$ .

Separate the terms.

Compare with the given form $a + \frac{b}{x+1}$ .

Try one yourself

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Express f(x) in the form a + b/(x+1), where a and b are constants. Hence, give a sequence of three transformations which take the graph of y = 1/x onto the graph of y = f(x). State the range of f. Form the composite function f(f(x)). Hence, or otherwise, obtain an expression for f⁻¹(x).

Answer

Step-by-step solution

Rewrite the numerator in terms of the denominator.

Substitute this back into the expression for f(x)f(x)f(x).

Separate the terms.

Compare with the given form a+bx+1a + \frac{b}{x+1}a+x+1b​.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Substitute this back into the expression for $f(x)$ .

Compare with the given form $a + \frac{b}{x+1}$ .