Find the inverse function f⁻¹ of the function f. Then, using a graphing utility, graph both f and f⁻¹ in the same viewing window. f(x)=7x+5.

The inverse function of f(x) = 7x + 5 is found by setting y = 7x + 5, solving for x to get f-1(x) = (x - 5)/7, and graphing both functions shows they are reflections across the line y = x.

Explanation:

To find the inverse function f -1 of the function f, we want to swap the roles of the x and y variables in the original function f(x) = 7x + 5 and solve for x in terms of y. This means we will set y = 7x + 5 and then solve for x. Here are the steps:

Set y = 7x + 5
Subtract 5 from both sides: y - 5 = 7x
Divide both sides by 7: (y - 5)/7 = x
The inverse function is then f-1(y) = (y - 5)/7. To express it as a function of x, we write f-1(x) = (x - 5)/7.

When using a graphing utility, you would graph both f(x) = 7x + 5 and its inverse f-1(x) = (x - 5)/7 in the same viewing window. The graph of f(x) is a straight line with a slope of 7 and a y-intercept of 5. The graph of f-1(x) will also be a straight line, but with a slope of 1/7 and a y-intercept of -5/7, and it will be a reflection of the graph of f(x) across the line y = x. According to the properties of functions and their inverses, the graph of an inverse function is always a reflection over the line y = x. This demonstrates the concept that f and f-1 are mirror images of each other across this diagonal.